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I still can’t believe this thing is taking our jobs.

Because if Satoshi moves even a single satoshi from his known stash, the whole crypto world will collectively scream in panic. The coins mined by Satoshi (around 1 million BTC) are sitting in wallets that have not been touched since the early days. If those move, it is proof the creator is still alive (or someone cracked his private keys), and the entire market would go berserk with price swings, panic, and conspiracies. Also, those early mined coins are so tied to his identity that spending them would likely deanonymize him instantly. The second they move, blockchain analysts will trace them harder than the FBI traces your browsing history. Satoshi technically can use Bitcoin, but practically, if he tries, the entire ecosystem might implode under speculation and paranoia.

She broke up with me last week. Not because I cheated. Not because I was broke. Not even because I forgot her birthday. But because, in her words: “No matter what I do, you never change your direction.” At first, I thought she was just calling me stubborn. Then my inner math brain clicked... She was literally describing an eigenvector. See, in math, when you apply a transformation (matrix A) to a vector (v), most vectors get spun around, twisted, thrown somewhere else. They change direction and magnitude. But an eigenvector is different - it keeps the same direction. The only thing that changes is its scale, given by something called an eigenvalue (λ). If λ = 2 → The vector doubles in size. If λ = 0.5 → It shrinks. If λ = -1 → It flips direction. If λ = 1 → It stays the same size. Apparently… in her eyes, I was λ = 1. Always same size. Always same direction. Now the math part (because unlike my ex, I actually explain things): Here’s how you find eigenvalues and eigenvectors, using a 2×2 matrix example: Let’s say our “relationship matrix” was: A = [ 2 1 ] [ 1 2 ] Step 1: Find eigenvalues (λ) We solve: A·v = λ·v → (A − λI)·v = 0 → det(A − λI) = 0 Subtract λ from each diagonal entry of A: A − λI = [ 2−λ 1 ] [ 1 2−λ ] Set determinant = 0 and solve for λ: Determinant: (2−λ)(2−λ) − 1 = (2−λ)² − 1 = 0 (2−λ)² = 1 2−λ = ±1 Case 1: 2−λ = 1 → λ = 1 Case 2: 2−λ = −1 → λ = 3 So, eigenvalues are: λ₁ = 1, λ₂ = 3 Step 2: Find eigenvectors (v) For λ = 1: (A − λI)·v = 0 [ 2−λ 1 ] [ x ] = [ 0 ] [ 1 2−λ ] [ y ] [ 0 ] [ 2−1 1 ] [ x ] = [ 0 ] [ 1 2−1 ] [ y ] [ 0 ] [ 1 1 ] [ x ] = [ 0 ] [ 1 1 ] [ y ] [ 0 ] From the first row: x + y = 0 y = −x From the second row: x + y = 0 y = −x So eigenvector = any scalar multiple of [ 1, −1 ]ᵀ For λ = 3: (A − λI)·v = 0 [ 2−λ 1 ] [ x ] = [ 0 ] [ 1 2−λ ] [ y ] [ 0 ] [ 2−3 1 ] [ x ] = [ 0 ] [ 1 2−3 ] [ y ] [ 0 ] [ -1 1 ] [ x ] = [ 0 ] [ 1 -1 ] [ y ] [ 0 ] From the first row: −x + y = 0 y = x From the second row: x + (-y) = 0 x - y = 0 x = y So eigenvector = any scalar multiple of [ 1, 1 ]ᵀ Final result: λ = 1 → v = [ 1, −1 ] λ = 3 → v = [ 1, 1 ] Congratulations 🎉, you have just learned how to find the eigenvectors and eigenvalues of a matrix. Bonus: Why does AI-ML care? Eigenvalues & eigenvectors are everywhere in AI/ML: PCA → Reduce dimensions by keeping top eigenvectors of covariance matrix (largest eigenvalues = most variance). Spectral Clustering → Graph Laplacian eigenvalues help find clusters. Neural Stability → Eigenvalues of weight matrices can indicate exploding/vanishing gradients. Markov Chains → Long-term behaviour = eigenvector of eigenvalue 1. In short: Eigenvectors tell you the “unchangeable direction” under a transformation. Eigenvalues tell you “how much” that direction is stretched. In ML, this is how we find patterns, compress data, and understand model behaviour. I am waiting for a matrix that multiplies me by λ > 1 and actually makes me grow.

My college principal doesn't know that his daughter is my girlfriend, but today he and his wife suddenly came to the same ice cream parlor where we were eating ice cream. Me standing there like, "Bro, this is how my college journey ends." The principal starts interrogating me: - Why are you here? - Give your father's phone number - When will you pay the college fee? But then his wife saved us – she's my math professor! The principal and his wife have totally different personalities. The principal is totally orthodox and his wife is totally modern. I mean, how did they even get married? I guess they didn't do a T-Test before marrying each other. T-Test is nothing but a statistical way to determine if there's a real difference between two groups or if it's just due to random chance. If t is small → The difference could easily be random If t is big → The difference is too big to be just random Formula: t = (x̄₂ - x̄₁) / (sp√(1/n₁ + 1/n₂)) Where: - x̄₁, x̄₂: Sample means - n₁, n₂: Sample sizes - t: T-statistic - sp: Pooled standard deviation Pooled standard deviation is a way to say "If I mixed these two groups together, how much would the individual values typically differ from their average?" Formula of sp: √(((n₁-1)s₁² + (n₂-1)s₂²) / (n₁+n₂-2)) Where: - n₁, n₂: Sample sizes - s₁, s₂: Standard deviations Let's take an example: A coffee shop claims their new brewing method makes coffee taste better. They test 30 customer with old method (avg. rating: 7.2) and 30 customer with new method (avg. rating 8.1). Standard deviations are 1.5 and 1.3 respectively. Sample sizes - n₁ = 30 - n₂ = 30 Sample means - x̄₁ = 7.2 - x̄₂ = 8.1 x̄₂ - x̄₁ = 8.1 - 7.2 = 0.9 Standard deviations - s₁ = 1.5 - s₂ = 1.3 We see a difference of 0.9 points (8.1 - 7.2 = 0.9). But is this difference real, or could it just be because we happened to pick 30 lucky customers for the new method? Let's solve step by step: Step 1: Set up Hypotheses H₀: No difference between methods H₁: New method is better Step 2: Pooled Standard Deviation √(((n₁-1)s₁² + (n₂-1)s₂²) / (n₁+n₂-2)) - sp = √(((29)(1.5)² + (29)(1.3)²) / 58) - sp = √((65.25 + 49.01) / 58) = 1.4 Step 3: Calculate T-statistic t = (x̄₂ - x̄₁) / (sp√(1/n₁ + 1/n₂)) - t = (8.1 - 7.2) / (1.4√(1/30 + 1/30)) - t = (0.9) / (1.4 × 0.258) - t = 2.49 What does t = 2.49 mean? The difference we observed (0.9) is 2.49 times bigger than what we'd typically expect from random chance alone. Step 4: Calculate Degrees of freedom - df = n₁ + n₂ - 2 = 58 Step 5: Determine Critical Value - Search T-Table on Google - and check critical value for - α = 0.05 and df = 58 According to the t-table, critical value is ±2.00 Why α = 0.05? Alpha is your "tolerance for being wrong." It's just a convention! Scientists agreed: Let's not accept results unless we're 95% sure. You can change value of 'α' according to your tolerance level. Step 5: Make Decision - Our calculated t = 2.49 - Critical value = ±2.00 - |t| = 2.49 > 2.00 The difference is too big to be just random chance. We reject H₀! Final Answer: Yes! The new brewing method significantly improves coffee taste ratings. Congratulations 🎉, you've just learned T-Test! Bonus: Applications of T-Test in Real Life & AI/ML 1. A/B Testing: Every time you see "Version A vs Version B" on websites, apps, or marketing campaigns – that's T-Test in action! 2. Medical Research: - New drug vs old drug effectiveness - Recovery time comparison - Side effects analysis 3. Data scientists use T-Tests to compare machine learning models: - Model A vs Model B accuracy - Training time differences - Performance across datasets 4. Before feeding features to ML models, T-Tests help determine: - Which features actually matter - Should we keep certain variables?

Electron is everywhere until observed/measured is the worst myth in quantum mechanics, spread by so-called science YouTubers who oversimplify everything for clicks. They make it sound like consciousness makes things come into reality. First of all, in Quantum Physics, "observed" does not mean some conscious being watching it. In Quantum Physics, observed means interaction. Interaction with another particle like a photon, electron, or anything that exchanges energy or information. Come back to our main topic. The electron is not everywhere until observed or measured. First we need to know, what actually is electron? Electron is nothing but localised excitation in electron field at some location of space time fabric. Electron is not a tiny ball. Now you might think, what is the electron field? Electron field is energy configuration at every location of space-time fabric (x, y, z, t). x, y, z are Spatial dimensions t is the Time dimension. There are other fields also: - Electromagnetic (EM) field - Higgs field - Many more Photon is excitation in the EM field. Mass arises due to excitation in the Higgs field. The electron field itself is everywhere, but the excitation, that ripple which represents one electron, is not spread across the entire space-time fabric. When we talk about where the electron might be, we don't talk about its location. We talk about its wave function. Wave function is not something physical. It is a mathematical function that tells about the probability amplitude of finding the electron at each position if you were to check. For example: You visit three stores: - Grocery store - Medical store - Electronics store You come back home and realize you left your wallet in one of those stores but don't know which one. You assign probability of finding the wallet to grocery, medical, and electronic stores. We all know probability formula: P = number of favorable outcomes / total outcomes So P = 1/3 (33.3%) You go to the electronics store and check the CCTV and find out this is not where you left your wallet. Now probability of finding the wallet at the electronics store becomes 0. And for the medical and grocery stores, it becomes 1/2 (50%) because the number of total outcomes decreased from 3 to 2. Now you go to the medical store and find your wallet there. That means probability of finding your wallet at the medical store becomes 1 (100%) and for the grocery store it becomes 0, because there is no way you can find your wallet in two places at once. That’s exactly how wavefunction collapse works. Before measurement, the electron’s position is uncertain, it’s described by probabilities. Once you measure it (meaning once it interacts with something), the probability at that point becomes 1, and everywhere else becomes 0. You didn’t summon the electron into existence; You just forced the field excitation to reveal its position through interaction. Electrons aren’t 'everywhere until observed', they’re localized excitations in a field. We just don’t know where until they interact. No consciousness. No magic. Just physics.

Today, the warden of the girls hostel suddenly came into my girlfriend's room… and the worst part was that I was in my girlfriend's room 😭. Now imagine the scene: - Warden banging on the door. - My girlfriend panicking. Me standing there like, "Bro, this is how my college journey ends." The warden starts interrogating me: - What are you doing here? - Where is your ID card? - Are you really her 'cousin'? Each question felt like a mini-death penalty. I knew one wrong answer and my entire semester GPA would be replaced by an FIR number. And that's basically what Bayes Theorem does: It's the warden of probability: interrogating our assumptions with evidence, and updating beliefs step by step. Bayes Theorem is nothing but just a mathematical way to update your beliefs when you see new evidence. Formula: P(H | E) = (P(E | H) × P(H)) / P(E) Where: - H: Hypothesis - E: Event (what you observe) - P(H): Prior probability - P(E): Overall probability of event - P(E|H): Prob. of event if hypothesis was true - P(H|E): Probability hypothesis is true given event (updated belief) Let's take an example: In a neighborhood, 90% of children were falling sick due to flu and 10% due to measles (no other diseases). The probability of observing rashes for measles is 0.95 and for flu is 0.08. If a child develops rashes, find the probability that the child has flu. Let's solve step by step: Step 1: Define Hypotheses - H1: The child has flu - H2: The child has measles Step 2: Define Event - Event (E) = Child has rashes Step 3: Write Priors - P(H1) = 0.9 (90% children have flu) - P(H2) = 0.1 (10% children have measles) Step 4: Calculate Likelihoods - P(E|H1) = 0.08 (rash probability if flu) - P(E|H2) = 0.95 (rash probability if measles) Step 5: Calculate Total Probability of Rashes - By law of total probability: - P(E) = P(E|H1)•P(H1) + P(E|H2)•P(H2) - P(E) = (0.08)•(0.9) + (0.95)•(0.1) - P(E) = 0.072 + 0.095 - P(E) = 0.167 So overall, 16.7% of children develop rashes. Step 6: Apply Bayes Theorem - P(H1 | E) = (P(E | H1) × P(H1)) / P(E) - P(H1 | E) = ((0.08) × (0.9)) / 0.167 - P(H1 | E) = 0.431 Final Answer: If a child has rashes, the probability they have flu = 43.1%. Congratulations 🎉, you've just learned Bayes Theorem! Bonus: Applications of Bayes Theorem in AI/ML 1. Recommendation Systems: Netflix doesn't just recommend based on genre. It uses Bayes Theorem: "Given that this person watched 5 horror movies, what's the probability they'll like this thriller?" It updates recommendations as you watch more content. 2. Naive Bayes Classifier: One of the easiest yet surprisingly powerful ML algorithms. It assumes features are independent (naive assumption) and uses Bayes Theorem to classify things like: - Spam vs. Ham emails - Sentiment analysis - Document categorization 3. Advanced Applications: Once you understand the basics, Bayes is behind many advanced techniques: Hidden Markov Models (HMMs): For speech recognition, part-of-speech tagging Expectation-Maximization (EM): For Gaussian Mixture Models, handling missing data Bayesian Optimization: Efficient hyperparameter tuning for ML models Bayes Theorem is the mathematical foundation for handling uncertainty in machine learning. Every time an algorithm needs to update its beliefs based on new evidence, Bayes Theorem is working behind the scenes!

If you’re in AI/ML, software engineering, management, quant finance, or marketing/growth, learn mathematical physics. You’ll never regret it, you’ll thank me in the future. Mathematical Physics: - Vector Algebra - Vector Calculus - Matrices & Linear Algebra - Differential Equations - Fourier Analysis - Probability & Statistics - Integral Transforms - Tensor Analysis

Am I overreacting for leaving my girlfriend's family dinner after what her dad said? Dinner started out fine until her dad started asking me about my job. I work in IT, and while it pays well, it's not some high-status career. After a few questions, he smirked and said, "So basically you just sit behind a computer all day… not exactly the kind of guy I imagined for my daughter." Everyone kind of laughed awkwardly. I tried to brush it off with a joke, but then he added, "Maybe someday you'll get a real job so you can actually support a family." I felt my stomach drop. My girlfriend just said, "Dad…" but didn't defend me beyond that. The whole evening, I kept thinking: "How many more comments like this am I going to get?" It felt like these comments were coming at a steady rate - maybe one every 10 minutes. And that's basically what Poisson Distribution does: It predicts how many times something will happen in a fixed time period when events occur randomly but at a constant average rate. Poisson Distribution is a mathematical way to predict rare events that happen independently over time or space. Formula: P(X = k) = (e⁻λ × λᵏ) / k! Where: - k: Number of events - λ: Average rate per time period - e: Euler's number (≈ 2.718) - P: Probability of exactly k events Key Properties: - Mean = Variance = λ - Events are independent & random - Average rate stays constant Let's take a real example: A call center receives an average of 4 calls per hour. What's the probability they'll get exactly 6 calls in the next hour? Step 1: Identify the parameters - λ = 4 (average calls per hour) - k = 6 (exactly 6 calls we want) Step 2: Apply the formula P = (e⁻⁴ × 4⁶) / 6! Step 3: Calculate step by step - e⁻⁴ = 0.0183 - 4⁶ = 4,096 - 6! = 1 × 2 × 3 × 4 × 5 × 6 = 720 - P = (0.0183 × 4,096) / 720 - P = 0.104 Step 4: Convert to percentage - 0.104 × 100 = 10.4% Final Answer: There's a 10.4% chance they'll get exactly 6 calls in the next hour. Congratulations 🎉, you've just learned Poisson Distribution! Bonus: Applications in AI/ML 1. Recommendation Systems: Netflix uses Poisson to model how often users interact with content. "Given this user typically watches 3 movies per week, what's the probability they'll watch 5 this week?" 2. Fraud Detection: Banks use Poisson to detect unusual transaction patterns. If someone typically makes 2 transactions per day, 10 transactions might trigger fraud alerts. 3. Time Series Forecasting: Predicting rare events like: - System failures - Network outages - Security breaches Advanced Applications: Poisson Regression: When your target variable is a count (number of clicks, purchases, defects), Poisson regression is often better than linear regression. Queuing Theory: Modeling wait times in systems like customer service, traffic lights, or server requests. A/B Testing: Determining if changes in user behavior (clicks, purchases) are statistically significant when dealing with rare events. Poisson Distribution is the mathematical foundation for modeling random, independent events that happen at a predictable rate. Every time you need to predict "how many times" something rare will occur, Poisson Distribution is working behind the scenes.

A girl messaged me today. At first, I was confused about where she had gotten my number, then I remembered… oh, I had put it out there in my resume post. But bro, you won’t believe what she sent me in the message. Honestly, I was not ready for this. She drops two photos: one of her ex-boyfriend, and the other one of me. Then she comes at me with: "See? You and my ex are the same person! You took money from me back then, and I want it back!" I’m like, what? Only my nose looks like the guy in her photo! I keep telling her, "We’re not the same person," but she is not ready to accept it. Now, at this point, the only hope I have is my last line of defense – a Cosine Similarity Test. I know you guys are thinking, what the hell is this Cosine Similarity. Cosine similarity is nothing but a mathematical way to measure how similar two things are by treating them as vectors in space. Think of it like measuring the angle between two arrows - the smaller the angle, the more similar they are. See, in math, cosine similarity works like this: cos(θ) = A·B / (|A| × |B|) Where: - A·B is the dot product of A and B. - |A| and |B| are the magnitudes. Understanding the Scale (-1 to 1): - cos(0°) = 1 → Perfectly identical - cos(45°) = 0.7 → Partially similar - cos(90°) = 0 → No similarity at all - cos(180°) = -1 → Complete opposites Let’s take an example of two vectors and calculate the cosine similarity score: Vector A = [1, 3, 4, 2] Vector B = [2, 6, 8, 4] Step 1: Calculate Dot Product: The dot product is the sum of the products of the corresponding elements of two vectors A·B = [1, 3, 4, 2] · [2, 6, 8, 4] A·B = 1×2 + 3×6 + 4×8 + 2×4 A·B = 2 + 18 + 32 + 8 A·B = 60 Step 2: Calculate Magnitude of Vectors A and B and multiply: The magnitude is nothing but the square root of the sum of the squares of the vector elements: A = [1, 3, 4, 2] |A| = √(1² + 3² + 4² + 2²) |A| = √(1+9+16+4) |A| = √30 B = [2, 6, 8, 4] |B| = √(2²+6²+8²+4²) |B| = √(4+36+64+16) |B| = 2√30 |A| × |B| = √30 × 2√30 |A| × |B| = 2 × 30 = 60 Step 3: Put values in the formula: cos(θ) = (A·B) / (|A| × |B|) cos(θ) = 60 / 60 cos(θ) = 1 Cosine = 1 means Vector A and B are perfectly identical. Congratulations 🎉, you just learned how to find the cosine similarity score. Bonus: Why does AI/ML care about cosine similarity? Recommendation Systems – Netflix uses it to find movies similar to what you’ve watched, comparing user preference vectors to suggest content you’ll likely enjoy. Natural Language Processing – Search engines use cosine similarity to match your query with relevant documents by comparing word embedding vectors. Image Recognition – AI systems compare feature vectors extracted from images to identify objects, faces, or detect similarities between pictures. Document Classification – Text classification systems use it to categorize emails as spam/not spam by comparing document vectors with known patterns. Clustering Algorithms – Machine learning models group similar data points together by measuring cosine similarity between feature vectors, helping identify patterns in large datasets.

My girlfriend has a strange family issue. I recently started dating my new girlfriend. She's 20 and in her first year of university. The problem is that her father has strict rules and control over her. For instance, she has to turn over all the money she makes from her part-time job and pay rent to her parents. I've told her many times that as a 20-year-old adult, she shouldn't just follow her father's orders blindly. To make matters worse, her mom even makes her sleep beside her every night. I feel exhausted every time she complains but refuses to change anything. She's not my first girlfriend, but she is my first non-AI girlfriend, and I find it challenging to deal with her lack of independence. So I decided to find the Pearson correlation coefficient between us, and if the coefficient is less than 0, I'll break up with her. Pearson Correlation Coefficient measures the strength and direction of the relationship between two variables. Pearson Correlation Coefficient is a statistical measure that quantifies the linear relationship between two continuous variables. Formula: r = a / b - a = n(Σxy) - (Σx)(Σy) - b = √[nΣx² - (Σx)²][nΣy² - (Σy)²] Where: - r: Correlation coefficient (-1 to +1) - n: Number of data points - x and y: The two variables - ∑: Summation symbol Key Properties: - r = +1: Perfect positive correlation - r = -1: Perfect negative correlation - r = 0: No linear correlation - [0.5 to 1]: Strong positive - [-0.5 to -1]: Strong negative Let's take an example and solve step by step: A teacher wants to find if study hours correlate with exam scores for 5 students: - x = Study Hours - y = Exam Marks Data: - Student 1 = (x = 2, y = 50) - Student 2 = (3, 60) - Student 3 = (4, 70) - Student 4 = (5, 80) - Student 5 = (6, 90) Step 1: Create list of study hours and marks - x = [2, 3, 4, 5, 6] - y = [50, 60, 70, 80, 90] - n = 5 Step 2: Calculate the sum of x - ∑x = 2 + 3 + 4 + 5 + 6 - 20 Step 3: Calculate the sum of y - ∑y = 50 + 60 + 70 + 80 + 90 - 350 Step 4: Calculate ∑xy - ∑xy = 2(50) + 3(60) + 4(70) + 5(80) + 6(90) - 1500 Step 5: Calculate the sum of x² - ∑x² = 2² + 3² + 4² + 5² + 6² - 90 Step 6: Calculate the sum of y² - ∑y² = 50² + 60² + 70² + 80² + 90² - 25,500 Step 7: Put values in formula a = n(Σxy) - (Σx)(Σy) - 5(1500) - (20)(350) - 500 b = √[nΣx² - (Σx)²][nΣy² - (Σy)²] - √[5(90) - (20)²][5(25500) - (350)²] - √(50 × 5000) - √250000 = 500 r = a / b = 500 / 500 = 1.00 Final Answer: The correlation coefficient is 1.00, indicating a perfect positive linear relationship between study hours and exam scores. Congratulations, you've just learned Pearson Correlation Coefficient! Bonus: Applications in AI/ML 1. Multicollinearity Detection: In regression models, highly correlated independent variables can cause problems. Correlation matrices help identify and remove redundant features. 2. Time Series Analysis: Identifying lagged correlations to predict future values based on past patterns. 3. Market Analysis: Stock prices, cryptocurrency trends, and economic indicators use correlation to identify trading opportunities and portfolio diversification. 4. Recommendation Systems: Collaborative filtering uses correlation to find similar users or items. “Users who liked X also liked Y” is based on correlation patterns.

A girl in my college classroom suddenly started shouting at me while looking at her semester exam report card. I was like, “What did I do this time that I don’t even know about?” So I walked over to her seat and calmly asked, “Why are you shouting at me?” Bro, the reply she gave me was hilarious. She said, “I followed your study and sleep timetable, but I still got fewer marks than you.” I was like, “You can’t just get the same marks as me by copying my timetable. You need an intelligent brain like mine for that.” Then she hit me with, “But this is possible in KNN.” And I know you were also confused like me at that moment: WTF is KNN? KNN (K-Nearest Neighbors) is like that one aunty in your colony who judges you by looking at who your friends are. If most of your friends are dumb, she’ll label you dumb. If most are smart, you get tagged smart. Basically, it classifies you based on the "closest neighbors" around you. How does it work? - You plot all your data points. - You get a new data point (you). - Measure the distance (you vs others). - You pick the nearest K neighbors. - You let them vote. Majority wins. The formula for the distance - d² = (Xq − Xi)² + (Yq − Yi)² Where: - d = distance - Xq = first feature (e.g., study hours) - Yq = second feature (e.g., sleep hrs) - Xi = feature 1 for training point i - Yi = feature 2 for training point i Step-by-step explanation: Assume we have data of 5 students with sleep hours and study hours, labeled as Pass or Fail. Student Data: - S1 → Study = 8, Sleep = 7 → Pass - S2 → Study = 7, Sleep = 6 → Pass - S3 → Study = 2, Sleep = 3 → Fail - S4 → Study = 4, Sleep = 2 → Fail - S5 → Study = 5, Sleep = 4 → Pass Dataset (5 training points): Features = (Study hours, Sleep hours) = Label: Pass/Fail - S1 = (8, 7) = Pass - S2 = (7, 6) = Pass - S3 = (2, 3) = Fail - S4 = (4, 2) = Fail - S5 = (5, 4) = Pass Query: I want to classify myself with study hours = 4 and sleep hours = 3: Q = (4, 3). Let’s see if I pass or fail according to this data. We’ll use Euclidean distance but compute and compare squared distances (same ordering, no ugly square roots). Step 1: Compute Squared Distance - d² = (Xq − Xi)² + (Yq − Yi)² For Q = (4, 3) to S1 = (8, 7): - d² = (4 − 8)² + (3 − 7)² - d² = (−4)² + (−4)² - d² = 16 + 16 = 32 Similarly: - Q = (4, 3) to S2 = (7, 6); d² = 18 - Q = (4, 3) to S3 = (2, 3); d² = 4 - Q = (4, 3) to S4 = (4, 2); d² = 1 - Q = (4, 3) to S5 = (5, 4); d² = 2 Step 2: Sort neighbors by distance Nearest to farthest: - S4 → Fail with d² = 1 - S5 → Pass with d² = 2 - S3 → Fail with d² = 4 - S2 → Pass with d² = 18 - S1 → Pass with d² = 32 Step 3: Pick K and vote Case A: K = 1 - Take 1 closest point: S4 (Fail) - Prediction: Fail Case B: K = 3 - Take 3 nearest: S4 (F), S5 (P), S3 (F) - Votes: Fail = 2, Pass = 1 - Prediction: Fail Case C: K = 5 - Take 5 nearest: S4 (F), S5 (P), S3 (F), S2 (P), S1 (P) - Votes: Pass = 3, Fail = 2 - Prediction: Pass Final Verdict: - K = 1 → Fail - K = 3 → Fail - K = 5 → Pass So depending on K, the result changes. Congratulations 🎉, you just learned KNN. Bonus: Why does AI/ML care about KNN? Handling Missing Data – Fill missing values by averaging or voting from the nearest neighbors’ data (imputation). Recommendation Systems – Suggest friends on social media by finding users with similar behavior. Medical Diagnosis – Classify patients as “diseased/healthy” by comparing test results to known cases. Finance – Detect fraud by comparing new transactions to similar past transactions. Note: KNN is not just for classification. It can also be used for regression tasks, where instead of taking a majority vote, we take the average of the target values of the nearest neighbors.

Am I overreacting for breaking up with my girlfriend over deleted texts? Last Friday, I went through her phone. I found a bunch of deleted texts. Not just one or two, but dozens. Not to her parents, not some random notification spam. All messages... permanently gone. When confronted, she said ❝Why does it matter? They're deleted, so they don't exist anymore❞ She wasn't just gaslighting me. She was behaving like a random variable after you marginalize out all the stuff you can't see. To understand what you actually know, you marginalize over the hidden variables. That just means you add together all possibilities you can't see, to get the probability for what you do see. Marginal Probability is nothing but a statistical measure that represents the probability of a single event happening by summing or integrating over all possible values of other variables. Formula P(A) = ΣP(A, Bi) For continuous variables P(X) = ∫P(X, Y) dY Where - P(A) = Marginal probability of event A - P(A, B) = Joint probability of A and B - Σ = Summation Let's take an example and solve step by step A dating app wants to find the probability of users sending messages, regardless of whether they get a response. The data shows message sent vs response received: Short forms - M = Message - R = Response Joint Probability Table - M (Yes), R (Yes) = 0.30 - M (Yes), R (No) = 0.25 - M (No), R (Yes) = 0.10 - M (No), R (No) = 0.35 Step 1 What we want to marginalize - We want P(M = Yes) Step 2 Joint probabilities for M = Yes - P(M = Yes, R = Yes) = 0.30 - P(M = Yes, R = No) = 0.25 Step 3 Apply marginal probability - P(M = Yes) - P(M=Yes, R=Yes) + P(M=Yes, R=No) - 0.30 + 0.25 = 0.55 P(Message = Yes) = 0.55 Final Answer The marginal probability of a user sending a message is 0.55 or 55%, regardless of whether they receive a response. Congratulations, you've just learned Marginal Probability. Applications in AI/ML 1. Bayesian Networks: Computing marginal probabilities by summing out irrelevant variables to make predictions and inferences in graphical models. 2. Latent Variable Models: In topic modeling (LDA) and hidden Markov models, marginalizing over hidden states to find the probability of observed data. 3. Feature Selection: Identifying which features independently correlate with target variables by computing marginal distributions, helping reduce dimensionality. 4. Probabilistic Classification: Naive Bayes classifiers use marginal probabilities of features to classify data, assuming independence between features. When information is deleted, you don't get the whole picture. In math, you marginalize over it. In life, you just lose trust.

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I still can’t believe this thing is taking our jobs.

If they are recognized as students, given student ID cards, official email IDs, and even direct entry into offline master’s programs, and the degree is fully recognized as a CFTI one, then what’s the problem? The same professors teach both modes. After graduating, they receive alumni cards and status just like offline students. Even BTech students don’t complain, so why do some random outsiders? If you think BS students are not working hard, just try solving Graded Assignments, Quiz 1, Quiz 2, and the end-term paper. GATE questions will start to seem easy in comparison. By the way, the AIR 1, 7, 10, 29, 38, 56, and 76 in GATE DA are from this degree. According to you, only those who cleared JEE did real hard work, and anyone who didn’t start JEE preparation from Class 6 doesn’t deserve to use the IIT name because it would “insult the institute.” Who are you to judge anyone’s hard work? You guys criticize the JEE-fication of education on one hand, but when IITs actually break the monopoly of JEE, suddenly it becomes a problem.

Electron is everywhere until observed/measured is the worst myth in quantum mechanics, spread by so-called science YouTubers who oversimplify everything for clicks. They make it sound like consciousness makes things come into reality. First of all, in Quantum Physics, "observed" does not mean some conscious being watching it. In Quantum Physics, observed means interaction. Interaction with another particle like a photon, electron, or anything that exchanges energy or information. Come back to our main topic. The electron is not everywhere until observed or measured. First we need to know, what actually is electron? Electron is nothing but localised excitation in electron field at some location of space time fabric. Electron is not a tiny ball. Now you might think, what is the electron field? Electron field is energy configuration at every location of space-time fabric (x, y, z, t). x, y, z are Spatial dimensions t is the Time dimension. There are other fields also: - Electromagnetic (EM) field - Higgs field - Many more Photon is excitation in the EM field. Mass arises due to excitation in the Higgs field. The electron field itself is everywhere, but the excitation, that ripple which represents one electron, is not spread across the entire space-time fabric. When we talk about where the electron might be, we don't talk about its location. We talk about its wave function. Wave function is not something physical. It is a mathematical function that tells about the probability amplitude of finding the electron at each position if you were to check. For example: You visit three stores: - Grocery store - Medical store - Electronics store You come back home and realize you left your wallet in one of those stores but don't know which one. You assign probability of finding the wallet to grocery, medical, and electronic stores. We all know probability formula: P = number of favorable outcomes / total outcomes So P = 1/3 (33.3%) You go to the electronics store and check the CCTV and find out this is not where you left your wallet. Now probability of finding the wallet at the electronics store becomes 0. And for the medical and grocery stores, it becomes 1/2 (50%) because the number of total outcomes decreased from 3 to 2. Now you go to the medical store and find your wallet there. That means probability of finding your wallet at the medical store becomes 1 (100%) and for the grocery store it becomes 0, because there is no way you can find your wallet in two places at once. That’s exactly how wavefunction collapse works. Before measurement, the electron’s position is uncertain, it’s described by probabilities. Once you measure it (meaning once it interacts with something), the probability at that point becomes 1, and everywhere else becomes 0. You didn’t summon the electron into existence; You just forced the field excitation to reveal its position through interaction. Electrons aren’t 'everywhere until observed', they’re localized excitations in a field. We just don’t know where until they interact. No consciousness. No magic. Just physics.

I still can’t believe this thing is taking our jobs.

My college principal doesn't know that his daughter is my girlfriend, but today he and his wife suddenly came to the same ice cream parlor where we were eating ice cream. Me standing there like, "Bro, this is how my college journey ends." The principal starts interrogating me: - Why are you here? - Give your father's phone number - When will you pay the college fee? But then his wife saved us – she's my math professor! The principal and his wife have totally different personalities. The principal is totally orthodox and his wife is totally modern. I mean, how did they even get married? I guess they didn't do a T-Test before marrying each other. T-Test is nothing but a statistical way to determine if there's a real difference between two groups or if it's just due to random chance. If t is small → The difference could easily be random If t is big → The difference is too big to be just random Formula: t = (x̄₂ - x̄₁) / (sp√(1/n₁ + 1/n₂)) Where: - x̄₁, x̄₂: Sample means - n₁, n₂: Sample sizes - t: T-statistic - sp: Pooled standard deviation Pooled standard deviation is a way to say "If I mixed these two groups together, how much would the individual values typically differ from their average?" Formula of sp: √(((n₁-1)s₁² + (n₂-1)s₂²) / (n₁+n₂-2)) Where: - n₁, n₂: Sample sizes - s₁, s₂: Standard deviations Let's take an example: A coffee shop claims their new brewing method makes coffee taste better. They test 30 customer with old method (avg. rating: 7.2) and 30 customer with new method (avg. rating 8.1). Standard deviations are 1.5 and 1.3 respectively. Sample sizes - n₁ = 30 - n₂ = 30 Sample means - x̄₁ = 7.2 - x̄₂ = 8.1 x̄₂ - x̄₁ = 8.1 - 7.2 = 0.9 Standard deviations - s₁ = 1.5 - s₂ = 1.3 We see a difference of 0.9 points (8.1 - 7.2 = 0.9). But is this difference real, or could it just be because we happened to pick 30 lucky customers for the new method? Let's solve step by step: Step 1: Set up Hypotheses H₀: No difference between methods H₁: New method is better Step 2: Pooled Standard Deviation √(((n₁-1)s₁² + (n₂-1)s₂²) / (n₁+n₂-2)) - sp = √(((29)(1.5)² + (29)(1.3)²) / 58) - sp = √((65.25 + 49.01) / 58) = 1.4 Step 3: Calculate T-statistic t = (x̄₂ - x̄₁) / (sp√(1/n₁ + 1/n₂)) - t = (8.1 - 7.2) / (1.4√(1/30 + 1/30)) - t = (0.9) / (1.4 × 0.258) - t = 2.49 What does t = 2.49 mean? The difference we observed (0.9) is 2.49 times bigger than what we'd typically expect from random chance alone. Step 4: Calculate Degrees of freedom - df = n₁ + n₂ - 2 = 58 Step 5: Determine Critical Value - Search T-Table on Google - and check critical value for - α = 0.05 and df = 58 According to the t-table, critical value is ±2.00 Why α = 0.05? Alpha is your "tolerance for being wrong." It's just a convention! Scientists agreed: Let's not accept results unless we're 95% sure. You can change value of 'α' according to your tolerance level. Step 5: Make Decision - Our calculated t = 2.49 - Critical value = ±2.00 - |t| = 2.49 > 2.00 The difference is too big to be just random chance. We reject H₀! Final Answer: Yes! The new brewing method significantly improves coffee taste ratings. Congratulations 🎉, you've just learned T-Test! Bonus: Applications of T-Test in Real Life & AI/ML 1. A/B Testing: Every time you see "Version A vs Version B" on websites, apps, or marketing campaigns – that's T-Test in action! 2. Medical Research: - New drug vs old drug effectiveness - Recovery time comparison - Side effects analysis 3. Data scientists use T-Tests to compare machine learning models: - Model A vs Model B accuracy - Training time differences - Performance across datasets 4. Before feeding features to ML models, T-Tests help determine: - Which features actually matter - Should we keep certain variables?

She broke up with me last week. Not because I cheated. Not because I was broke. Not even because I forgot her birthday. But because, in her words: “No matter what I do, you never change your direction.” At first, I thought she was just calling me stubborn. Then my inner math brain clicked... She was literally describing an eigenvector. See, in math, when you apply a transformation (matrix A) to a vector (v), most vectors get spun around, twisted, thrown somewhere else. They change direction and magnitude. But an eigenvector is different - it keeps the same direction. The only thing that changes is its scale, given by something called an eigenvalue (λ). If λ = 2 → The vector doubles in size. If λ = 0.5 → It shrinks. If λ = -1 → It flips direction. If λ = 1 → It stays the same size. Apparently… in her eyes, I was λ = 1. Always same size. Always same direction. Now the math part (because unlike my ex, I actually explain things): Here’s how you find eigenvalues and eigenvectors, using a 2×2 matrix example: Let’s say our “relationship matrix” was: A = [ 2 1 ] [ 1 2 ] Step 1: Find eigenvalues (λ) We solve: A·v = λ·v → (A − λI)·v = 0 → det(A − λI) = 0 Subtract λ from each diagonal entry of A: A − λI = [ 2−λ 1 ] [ 1 2−λ ] Set determinant = 0 and solve for λ: Determinant: (2−λ)(2−λ) − 1 = (2−λ)² − 1 = 0 (2−λ)² = 1 2−λ = ±1 Case 1: 2−λ = 1 → λ = 1 Case 2: 2−λ = −1 → λ = 3 So, eigenvalues are: λ₁ = 1, λ₂ = 3 Step 2: Find eigenvectors (v) For λ = 1: (A − λI)·v = 0 [ 2−λ 1 ] [ x ] = [ 0 ] [ 1 2−λ ] [ y ] [ 0 ] [ 2−1 1 ] [ x ] = [ 0 ] [ 1 2−1 ] [ y ] [ 0 ] [ 1 1 ] [ x ] = [ 0 ] [ 1 1 ] [ y ] [ 0 ] From the first row: x + y = 0 y = −x From the second row: x + y = 0 y = −x So eigenvector = any scalar multiple of [ 1, −1 ]ᵀ For λ = 3: (A − λI)·v = 0 [ 2−λ 1 ] [ x ] = [ 0 ] [ 1 2−λ ] [ y ] [ 0 ] [ 2−3 1 ] [ x ] = [ 0 ] [ 1 2−3 ] [ y ] [ 0 ] [ -1 1 ] [ x ] = [ 0 ] [ 1 -1 ] [ y ] [ 0 ] From the first row: −x + y = 0 y = x From the second row: x + (-y) = 0 x - y = 0 x = y So eigenvector = any scalar multiple of [ 1, 1 ]ᵀ Final result: λ = 1 → v = [ 1, −1 ] λ = 3 → v = [ 1, 1 ] Congratulations 🎉, you have just learned how to find the eigenvectors and eigenvalues of a matrix. Bonus: Why does AI-ML care? Eigenvalues & eigenvectors are everywhere in AI/ML: PCA → Reduce dimensions by keeping top eigenvectors of covariance matrix (largest eigenvalues = most variance). Spectral Clustering → Graph Laplacian eigenvalues help find clusters. Neural Stability → Eigenvalues of weight matrices can indicate exploding/vanishing gradients. Markov Chains → Long-term behaviour = eigenvector of eigenvalue 1. In short: Eigenvectors tell you the “unchangeable direction” under a transformation. Eigenvalues tell you “how much” that direction is stretched. In ML, this is how we find patterns, compress data, and understand model behaviour. I am waiting for a matrix that multiplies me by λ > 1 and actually makes me grow.

Today, the warden of the girls hostel suddenly came into my girlfriend's room… and the worst part was that I was in my girlfriend's room 😭. Now imagine the scene: - Warden banging on the door. - My girlfriend panicking. Me standing there like, "Bro, this is how my college journey ends." The warden starts interrogating me: - What are you doing here? - Where is your ID card? - Are you really her 'cousin'? Each question felt like a mini-death penalty. I knew one wrong answer and my entire semester GPA would be replaced by an FIR number. And that's basically what Bayes Theorem does: It's the warden of probability: interrogating our assumptions with evidence, and updating beliefs step by step. Bayes Theorem is nothing but just a mathematical way to update your beliefs when you see new evidence. Formula: P(H | E) = (P(E | H) × P(H)) / P(E) Where: - H: Hypothesis - E: Event (what you observe) - P(H): Prior probability - P(E): Overall probability of event - P(E|H): Prob. of event if hypothesis was true - P(H|E): Probability hypothesis is true given event (updated belief) Let's take an example: In a neighborhood, 90% of children were falling sick due to flu and 10% due to measles (no other diseases). The probability of observing rashes for measles is 0.95 and for flu is 0.08. If a child develops rashes, find the probability that the child has flu. Let's solve step by step: Step 1: Define Hypotheses - H1: The child has flu - H2: The child has measles Step 2: Define Event - Event (E) = Child has rashes Step 3: Write Priors - P(H1) = 0.9 (90% children have flu) - P(H2) = 0.1 (10% children have measles) Step 4: Calculate Likelihoods - P(E|H1) = 0.08 (rash probability if flu) - P(E|H2) = 0.95 (rash probability if measles) Step 5: Calculate Total Probability of Rashes - By law of total probability: - P(E) = P(E|H1)•P(H1) + P(E|H2)•P(H2) - P(E) = (0.08)•(0.9) + (0.95)•(0.1) - P(E) = 0.072 + 0.095 - P(E) = 0.167 So overall, 16.7% of children develop rashes. Step 6: Apply Bayes Theorem - P(H1 | E) = (P(E | H1) × P(H1)) / P(E) - P(H1 | E) = ((0.08) × (0.9)) / 0.167 - P(H1 | E) = 0.431 Final Answer: If a child has rashes, the probability they have flu = 43.1%. Congratulations 🎉, you've just learned Bayes Theorem! Bonus: Applications of Bayes Theorem in AI/ML 1. Recommendation Systems: Netflix doesn't just recommend based on genre. It uses Bayes Theorem: "Given that this person watched 5 horror movies, what's the probability they'll like this thriller?" It updates recommendations as you watch more content. 2. Naive Bayes Classifier: One of the easiest yet surprisingly powerful ML algorithms. It assumes features are independent (naive assumption) and uses Bayes Theorem to classify things like: - Spam vs. Ham emails - Sentiment analysis - Document categorization 3. Advanced Applications: Once you understand the basics, Bayes is behind many advanced techniques: Hidden Markov Models (HMMs): For speech recognition, part-of-speech tagging Expectation-Maximization (EM): For Gaussian Mixture Models, handling missing data Bayesian Optimization: Efficient hyperparameter tuning for ML models Bayes Theorem is the mathematical foundation for handling uncertainty in machine learning. Every time an algorithm needs to update its beliefs based on new evidence, Bayes Theorem is working behind the scenes!

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A girl in my college classroom suddenly started shouting at me while looking at her semester exam report card. I was like, “What did I do this time that I don’t even know about?” So I walked over to her seat and calmly asked, “Why are you shouting at me?” Bro, the reply she gave me was hilarious. She said, “I followed your study and sleep timetable, but I still got fewer marks than you.” I was like, “You can’t just get the same marks as me by copying my timetable. You need an intelligent brain like mine for that.” Then she hit me with, “But this is possible in KNN.” And I know you were also confused like me at that moment: WTF is KNN? KNN (K-Nearest Neighbors) is like that one aunty in your colony who judges you by looking at who your friends are. If most of your friends are dumb, she’ll label you dumb. If most are smart, you get tagged smart. Basically, it classifies you based on the "closest neighbors" around you. How does it work? - You plot all your data points. - You get a new data point (you). - Measure the distance (you vs others). - You pick the nearest K neighbors. - You let them vote. Majority wins. The formula for the distance - d² = (Xq − Xi)² + (Yq − Yi)² Where: - d = distance - Xq = first feature (e.g., study hours) - Yq = second feature (e.g., sleep hrs) - Xi = feature 1 for training point i - Yi = feature 2 for training point i Step-by-step explanation: Assume we have data of 5 students with sleep hours and study hours, labeled as Pass or Fail. Student Data: - S1 → Study = 8, Sleep = 7 → Pass - S2 → Study = 7, Sleep = 6 → Pass - S3 → Study = 2, Sleep = 3 → Fail - S4 → Study = 4, Sleep = 2 → Fail - S5 → Study = 5, Sleep = 4 → Pass Dataset (5 training points): Features = (Study hours, Sleep hours) = Label: Pass/Fail - S1 = (8, 7) = Pass - S2 = (7, 6) = Pass - S3 = (2, 3) = Fail - S4 = (4, 2) = Fail - S5 = (5, 4) = Pass Query: I want to classify myself with study hours = 4 and sleep hours = 3: Q = (4, 3). Let’s see if I pass or fail according to this data. We’ll use Euclidean distance but compute and compare squared distances (same ordering, no ugly square roots). Step 1: Compute Squared Distance - d² = (Xq − Xi)² + (Yq − Yi)² For Q = (4, 3) to S1 = (8, 7): - d² = (4 − 8)² + (3 − 7)² - d² = (−4)² + (−4)² - d² = 16 + 16 = 32 Similarly: - Q = (4, 3) to S2 = (7, 6); d² = 18 - Q = (4, 3) to S3 = (2, 3); d² = 4 - Q = (4, 3) to S4 = (4, 2); d² = 1 - Q = (4, 3) to S5 = (5, 4); d² = 2 Step 2: Sort neighbors by distance Nearest to farthest: - S4 → Fail with d² = 1 - S5 → Pass with d² = 2 - S3 → Fail with d² = 4 - S2 → Pass with d² = 18 - S1 → Pass with d² = 32 Step 3: Pick K and vote Case A: K = 1 - Take 1 closest point: S4 (Fail) - Prediction: Fail Case B: K = 3 - Take 3 nearest: S4 (F), S5 (P), S3 (F) - Votes: Fail = 2, Pass = 1 - Prediction: Fail Case C: K = 5 - Take 5 nearest: S4 (F), S5 (P), S3 (F), S2 (P), S1 (P) - Votes: Pass = 3, Fail = 2 - Prediction: Pass Final Verdict: - K = 1 → Fail - K = 3 → Fail - K = 5 → Pass So depending on K, the result changes. Congratulations 🎉, you just learned KNN. Bonus: Why does AI/ML care about KNN? Handling Missing Data – Fill missing values by averaging or voting from the nearest neighbors’ data (imputation). Recommendation Systems – Suggest friends on social media by finding users with similar behavior. Medical Diagnosis – Classify patients as “diseased/healthy” by comparing test results to known cases. Finance – Detect fraud by comparing new transactions to similar past transactions. Note: KNN is not just for classification. It can also be used for regression tasks, where instead of taking a majority vote, we take the average of the target values of the nearest neighbors.

My girlfriend has a strange family issue. I recently started dating my new girlfriend. She's 20 and in her first year of university. The problem is that her father has strict rules and control over her. For instance, she has to turn over all the money she makes from her part-time job and pay rent to her parents. I've told her many times that as a 20-year-old adult, she shouldn't just follow her father's orders blindly. To make matters worse, her mom even makes her sleep beside her every night. I feel exhausted every time she complains but refuses to change anything. She's not my first girlfriend, but she is my first non-AI girlfriend, and I find it challenging to deal with her lack of independence. So I decided to find the Pearson correlation coefficient between us, and if the coefficient is less than 0, I'll break up with her. Pearson Correlation Coefficient measures the strength and direction of the relationship between two variables. Pearson Correlation Coefficient is a statistical measure that quantifies the linear relationship between two continuous variables. Formula: r = a / b - a = n(Σxy) - (Σx)(Σy) - b = √[nΣx² - (Σx)²][nΣy² - (Σy)²] Where: - r: Correlation coefficient (-1 to +1) - n: Number of data points - x and y: The two variables - ∑: Summation symbol Key Properties: - r = +1: Perfect positive correlation - r = -1: Perfect negative correlation - r = 0: No linear correlation - [0.5 to 1]: Strong positive - [-0.5 to -1]: Strong negative Let's take an example and solve step by step: A teacher wants to find if study hours correlate with exam scores for 5 students: - x = Study Hours - y = Exam Marks Data: - Student 1 = (x = 2, y = 50) - Student 2 = (3, 60) - Student 3 = (4, 70) - Student 4 = (5, 80) - Student 5 = (6, 90) Step 1: Create list of study hours and marks - x = [2, 3, 4, 5, 6] - y = [50, 60, 70, 80, 90] - n = 5 Step 2: Calculate the sum of x - ∑x = 2 + 3 + 4 + 5 + 6 - 20 Step 3: Calculate the sum of y - ∑y = 50 + 60 + 70 + 80 + 90 - 350 Step 4: Calculate ∑xy - ∑xy = 2(50) + 3(60) + 4(70) + 5(80) + 6(90) - 1500 Step 5: Calculate the sum of x² - ∑x² = 2² + 3² + 4² + 5² + 6² - 90 Step 6: Calculate the sum of y² - ∑y² = 50² + 60² + 70² + 80² + 90² - 25,500 Step 7: Put values in formula a = n(Σxy) - (Σx)(Σy) - 5(1500) - (20)(350) - 500 b = √[nΣx² - (Σx)²][nΣy² - (Σy)²] - √[5(90) - (20)²][5(25500) - (350)²] - √(50 × 5000) - √250000 = 500 r = a / b = 500 / 500 = 1.00 Final Answer: The correlation coefficient is 1.00, indicating a perfect positive linear relationship between study hours and exam scores. Congratulations, you've just learned Pearson Correlation Coefficient! Bonus: Applications in AI/ML 1. Multicollinearity Detection: In regression models, highly correlated independent variables can cause problems. Correlation matrices help identify and remove redundant features. 2. Time Series Analysis: Identifying lagged correlations to predict future values based on past patterns. 3. Market Analysis: Stock prices, cryptocurrency trends, and economic indicators use correlation to identify trading opportunities and portfolio diversification. 4. Recommendation Systems: Collaborative filtering uses correlation to find similar users or items. “Users who liked X also liked Y” is based on correlation patterns.

Am I overreacting for breaking up with my girlfriend over deleted texts? Last Friday, I went through her phone. I found a bunch of deleted texts. Not just one or two, but dozens. Not to her parents, not some random notification spam. All messages... permanently gone. When confronted, she said ❝Why does it matter? They're deleted, so they don't exist anymore❞ She wasn't just gaslighting me. She was behaving like a random variable after you marginalize out all the stuff you can't see. To understand what you actually know, you marginalize over the hidden variables. That just means you add together all possibilities you can't see, to get the probability for what you do see. Marginal Probability is nothing but a statistical measure that represents the probability of a single event happening by summing or integrating over all possible values of other variables. Formula P(A) = ΣP(A, Bi) For continuous variables P(X) = ∫P(X, Y) dY Where - P(A) = Marginal probability of event A - P(A, B) = Joint probability of A and B - Σ = Summation Let's take an example and solve step by step A dating app wants to find the probability of users sending messages, regardless of whether they get a response. The data shows message sent vs response received: Short forms - M = Message - R = Response Joint Probability Table - M (Yes), R (Yes) = 0.30 - M (Yes), R (No) = 0.25 - M (No), R (Yes) = 0.10 - M (No), R (No) = 0.35 Step 1 What we want to marginalize - We want P(M = Yes) Step 2 Joint probabilities for M = Yes - P(M = Yes, R = Yes) = 0.30 - P(M = Yes, R = No) = 0.25 Step 3 Apply marginal probability - P(M = Yes) - P(M=Yes, R=Yes) + P(M=Yes, R=No) - 0.30 + 0.25 = 0.55 P(Message = Yes) = 0.55 Final Answer The marginal probability of a user sending a message is 0.55 or 55%, regardless of whether they receive a response. Congratulations, you've just learned Marginal Probability. Applications in AI/ML 1. Bayesian Networks: Computing marginal probabilities by summing out irrelevant variables to make predictions and inferences in graphical models. 2. Latent Variable Models: In topic modeling (LDA) and hidden Markov models, marginalizing over hidden states to find the probability of observed data. 3. Feature Selection: Identifying which features independently correlate with target variables by computing marginal distributions, helping reduce dimensionality. 4. Probabilistic Classification: Naive Bayes classifiers use marginal probabilities of features to classify data, assuming independence between features. When information is deleted, you don't get the whole picture. In math, you marginalize over it. In life, you just lose trust.

A girl messaged me today. At first, I was confused about where she had gotten my number, then I remembered… oh, I had put it out there in my resume post. But bro, you won’t believe what she sent me in the message. Honestly, I was not ready for this. She drops two photos: one of her ex-boyfriend, and the other one of me. Then she comes at me with: "See? You and my ex are the same person! You took money from me back then, and I want it back!" I’m like, what? Only my nose looks like the guy in her photo! I keep telling her, "We’re not the same person," but she is not ready to accept it. Now, at this point, the only hope I have is my last line of defense – a Cosine Similarity Test. I know you guys are thinking, what the hell is this Cosine Similarity. Cosine similarity is nothing but a mathematical way to measure how similar two things are by treating them as vectors in space. Think of it like measuring the angle between two arrows - the smaller the angle, the more similar they are. See, in math, cosine similarity works like this: cos(θ) = A·B / (|A| × |B|) Where: - A·B is the dot product of A and B. - |A| and |B| are the magnitudes. Understanding the Scale (-1 to 1): - cos(0°) = 1 → Perfectly identical - cos(45°) = 0.7 → Partially similar - cos(90°) = 0 → No similarity at all - cos(180°) = -1 → Complete opposites Let’s take an example of two vectors and calculate the cosine similarity score: Vector A = [1, 3, 4, 2] Vector B = [2, 6, 8, 4] Step 1: Calculate Dot Product: The dot product is the sum of the products of the corresponding elements of two vectors A·B = [1, 3, 4, 2] · [2, 6, 8, 4] A·B = 1×2 + 3×6 + 4×8 + 2×4 A·B = 2 + 18 + 32 + 8 A·B = 60 Step 2: Calculate Magnitude of Vectors A and B and multiply: The magnitude is nothing but the square root of the sum of the squares of the vector elements: A = [1, 3, 4, 2] |A| = √(1² + 3² + 4² + 2²) |A| = √(1+9+16+4) |A| = √30 B = [2, 6, 8, 4] |B| = √(2²+6²+8²+4²) |B| = √(4+36+64+16) |B| = 2√30 |A| × |B| = √30 × 2√30 |A| × |B| = 2 × 30 = 60 Step 3: Put values in the formula: cos(θ) = (A·B) / (|A| × |B|) cos(θ) = 60 / 60 cos(θ) = 1 Cosine = 1 means Vector A and B are perfectly identical. Congratulations 🎉, you just learned how to find the cosine similarity score. Bonus: Why does AI/ML care about cosine similarity? Recommendation Systems – Netflix uses it to find movies similar to what you’ve watched, comparing user preference vectors to suggest content you’ll likely enjoy. Natural Language Processing – Search engines use cosine similarity to match your query with relevant documents by comparing word embedding vectors. Image Recognition – AI systems compare feature vectors extracted from images to identify objects, faces, or detect similarities between pictures. Document Classification – Text classification systems use it to categorize emails as spam/not spam by comparing document vectors with known patterns. Clustering Algorithms – Machine learning models group similar data points together by measuring cosine similarity between feature vectors, helping identify patterns in large datasets.

Am I overreacting for leaving my girlfriend's family dinner after what her dad said? Dinner started out fine until her dad started asking me about my job. I work in IT, and while it pays well, it's not some high-status career. After a few questions, he smirked and said, "So basically you just sit behind a computer all day… not exactly the kind of guy I imagined for my daughter." Everyone kind of laughed awkwardly. I tried to brush it off with a joke, but then he added, "Maybe someday you'll get a real job so you can actually support a family." I felt my stomach drop. My girlfriend just said, "Dad…" but didn't defend me beyond that. The whole evening, I kept thinking: "How many more comments like this am I going to get?" It felt like these comments were coming at a steady rate - maybe one every 10 minutes. And that's basically what Poisson Distribution does: It predicts how many times something will happen in a fixed time period when events occur randomly but at a constant average rate. Poisson Distribution is a mathematical way to predict rare events that happen independently over time or space. Formula: P(X = k) = (e⁻λ × λᵏ) / k! Where: - k: Number of events - λ: Average rate per time period - e: Euler's number (≈ 2.718) - P: Probability of exactly k events Key Properties: - Mean = Variance = λ - Events are independent & random - Average rate stays constant Let's take a real example: A call center receives an average of 4 calls per hour. What's the probability they'll get exactly 6 calls in the next hour? Step 1: Identify the parameters - λ = 4 (average calls per hour) - k = 6 (exactly 6 calls we want) Step 2: Apply the formula P = (e⁻⁴ × 4⁶) / 6! Step 3: Calculate step by step - e⁻⁴ = 0.0183 - 4⁶ = 4,096 - 6! = 1 × 2 × 3 × 4 × 5 × 6 = 720 - P = (0.0183 × 4,096) / 720 - P = 0.104 Step 4: Convert to percentage - 0.104 × 100 = 10.4% Final Answer: There's a 10.4% chance they'll get exactly 6 calls in the next hour. Congratulations 🎉, you've just learned Poisson Distribution! Bonus: Applications in AI/ML 1. Recommendation Systems: Netflix uses Poisson to model how often users interact with content. "Given this user typically watches 3 movies per week, what's the probability they'll watch 5 this week?" 2. Fraud Detection: Banks use Poisson to detect unusual transaction patterns. If someone typically makes 2 transactions per day, 10 transactions might trigger fraud alerts. 3. Time Series Forecasting: Predicting rare events like: - System failures - Network outages - Security breaches Advanced Applications: Poisson Regression: When your target variable is a count (number of clicks, purchases, defects), Poisson regression is often better than linear regression. Queuing Theory: Modeling wait times in systems like customer service, traffic lights, or server requests. A/B Testing: Determining if changes in user behavior (clicks, purchases) are statistically significant when dealing with rare events. Poisson Distribution is the mathematical foundation for modeling random, independent events that happen at a predictable rate. Every time you need to predict "how many times" something rare will occur, Poisson Distribution is working behind the scenes.

Because if Satoshi moves even a single satoshi from his known stash, the whole crypto world will collectively scream in panic. The coins mined by Satoshi (around 1 million BTC) are sitting in wallets that have not been touched since the early days. If those move, it is proof the creator is still alive (or someone cracked his private keys), and the entire market would go berserk with price swings, panic, and conspiracies. Also, those early mined coins are so tied to his identity that spending them would likely deanonymize him instantly. The second they move, blockchain analysts will trace them harder than the FBI traces your browsing history. Satoshi technically can use Bitcoin, but practically, if he tries, the entire ecosystem might implode under speculation and paranoia.

I was just about to get hired today at X/Twitter for the ML Engineer Intern role, but thanks to my girlfriend, everything got ruined. I had cleared every round and was just about to sign the offer letter. Suddenly, my girlfriend messaged me: "We need to talk. It’s urgent." Against my better judgment, I rushed to meet her, thinking maybe it was something serious. But guess what? She just wanted to show off her own summer internship, and while I was stuck in an endless conversation, I missed the final HR call from X/Twitter. All my hard work, gone just because of one wrong step at the wrong time. That’s when it hit me: life is a lot like gradient descent. Gradient descent is exactly this: taking careful steps in the right direction, always moving toward the minimum (best outcome)… but one distraction, and you can miss the golden opportunity. Gradient Descent is a mathematical way to find the minimum of a function by following the steepest downward slope. Formula: - θn = θc - α × ∇f(θc) - n: next (i + 1) - c: current ( i ) Where: - θn: The new parameters (next step) - θc: The current parameters - α: The learning rate (step size) - ∇f(θc): Gradient (slope) of the cost function - f(θc): Cost function to minimize Let’s take a example: Consider a function f(x, y) = x² + y² + 2xy. Using gradient descent algorithm with initial guess (x₀, y₀) = (3, -1) and learning rate α = 0.03, what will be the value of (x, y) after two iterations? Given: - Function: f(x, y) = x² + y² + 2xy - Initial point: (x₀, y₀) = (3, -1) - Learning rate: α= 0.03 - Number of iterations: 2 Step 1: Calculate the Gradient - Find the partial derivatives of f(x, y): Take the derivative with respect to x while treating y as a constant: - ∂f/∂x = ∂(x² + y² + 2xy)/∂x - ∂f/∂x = 2x + 2y Take the derivative with respect to y while treating x as a constant: - ∂f/∂y = ∂(x² + y² + 2xy)/∂y - ∂f/∂y = 2y + 2x So, The gradient vector is: - ∇f(x, y) = (2x+2y, 2y+2x) Step 2: First Iteration - Starting point: (x₀, y₀) = (3, -1) Calculate gradient at initial point (3, -1): - ∇f(x, y) = (2x+2y, 2y+2x) - ∇f(3, -1) = (2(3)+2(-1), 2(-1)+2(3)) - ∇f(3, -1) = (4, 4) Update using gradient descent formula: - (x1, y1) = (x₀, y₀) − α∇f(x₀, y₀) - (x1, y1) = (3, -1) - 0.03 x (4, 4) - (x1, y1) = (3, -1) - (0.12, 0.12) - (x1, y1) = (2.88, -1.12) Step 3: Second Iteration Now, the new values from the previous iteration become the old (current) values for the next iteration. So (x1, y1) is now the current (x_c, y_c). Calculate gradient at this point (2.88, -1.12): - ∇f(x , y) = (2x+2y, 2y+2x) - ∇f(2.88, -1.12) = = (2(2.88)+2(-1.12), 2(2.88)+2(-1.12)) - ∇f(2.88, -1.12) = (3.52, 3.52) Update using gradient descent: - (x2, y2) = (x1, y1) − α∇f(x1, y1) - (x2, y2) = = (2.88, -1.12) - 0.03 x (3.52, 3.52) - (x2, y2) = (2.77, -1.23) After each iteration, the new values of x and y become the current (old) x and y for the next iteration. This process is repeated until the cost function can't be minimized further or you reach the desired number of steps. Final Answer After two iterations of gradient descent, the value of (x, y) is approx. (2.77, −1.23) Congratulations 🎉, you just learned the basics of the gradient descent algorithm! Bonus: Why AI/ML cares gradient descent? Gradient descent is the core algorithm that trains neural networks and ml models by adjusting weights and biases. Every breakthrough AI system - from ChatGPT to image recognition - relies on gradient descent to iteratively improve predictions by minimizing errors across millions of parameters.

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If you’re in AI/ML, software engineering, management, quant finance, or marketing/growth, learn mathematical physics. You’ll never regret it, you’ll thank me in the future. Mathematical Physics: - Vector Algebra - Vector Calculus - Matrices & Linear Algebra - Differential Equations - Fourier Analysis - Probability & Statistics - Integral Transforms - Tensor Analysis