12. Expectation, Variance, Covariance & Pearson Correlation Coefficient • Mathematics, Statistics & Machine Learning (Big Tech Prep) • Rishabh Ki Kaksha

Mathematics, Statistics & Machine Learning (Big Tech Prep) › 12. Expectation, Variance, Covariance & Pearson Correlation Coefficient

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Course Syllabus

Module 1: Linear Algebra for Machine Learning & Deep Learning

01. Vectors, Vector Spaces, Dot Products & Norms (L1, L2, L-Infinity) in High Dimensions 02. Matrices, Matrix Multiplication & Linear Transformations (Rotation, Scaling, Shear) 03. Eigenvalues & Eigenvectors: Geometric Intuition, Diagonalization & Spectral Theorem 04. Principal Component Analysis (PCA) & Singular Value Decomposition (SVD) Math

Module 2: Multivariable Calculus, Gradients & Optimization Algorithms

05. Partial Derivatives, Directional Derivatives & The Gradient Vector (Del Operator) 06. The Jacobian Matrix & Hessian Matrix (Second Derivative Curvature Analysis) 07. Multivariate Chain Rule & The Mathematical Backbone of Neural Backpropagation 08. Optimization Algorithms: Gradient Descent, SGD with Momentum, RMSprop & Adam Optimizer Math

Module 3: Probability Theory & Random Variables

09. Probability Foundations: Axioms of Probability, Conditional Probability & Independence 10. Bayes' Theorem & Bayesian Inference: Prior, Likelihood, Posterior & Evidence 11. Probability Distributions: Gaussian (Normal), Bernoulli, Binomial, Poisson & Exponential Distributions

Module 4: Inferential Statistics & Hypothesis Testing

12. Expectation, Variance, Covariance & Pearson Correlation Coefficient 13. Law of Large Numbers & Central Limit Theorem (CLT) with Monte Carlo Simulations 14. Hypothesis Testing: Null Hypothesis, p-Values, Z-Test, Student's t-Test & Chi-Square Test 15. Maximum Likelihood Estimation (MLE) vs Maximum A Posteriori (MAP) Estimation

Module 5: Classical Machine Learning Algorithms from Scratch (NumPy)

16. Linear Regression from Scratch: Ordinary Least Squares (OLS) Closed Form vs Gradient Descent 17. Logistic Regression & Binary Cross-Entropy Loss from Scratch with NumPy 18. Regularization Mathematics: L1 Lasso (Sparsity) vs L2 Ridge (Weight Decay) Penalties

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# 12. Expectation, Variance, Covariance & Pearson Correlation Coefficient

> Mathematics & ML Foundations Notes by Rishabh - Topic 12

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### 1. Mathematical Formulas & Intuition 📐

*   **Expected Value (Mean):**
    $$\mathbb{E}[X] = \sum x P(X=x) \quad \text{or} \quad \int x f(x) dx$$
*   **Variance (Spread of data around mean):**
    $$\text{Var}(X) = \mathbb{E}[(X - \mathbb{E}[X])^2] = \mathbb{E}[X^2] - (\mathbb{E}[X])^2$$
*   **Covariance (Joint variability of 2 variables):**
    $$\text{Cov}(X, Y) = \mathbb{E}[(X - \mu_X)(Y - \mu_Y)] = \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y]$$
*   **Pearson Correlation Coefficient ($\rho \in [-1, 1]$):**
    $$\rho_{X, Y} = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y}$$

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### 2. Master Comparison Matrix

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### 3. Interview Ke 3 Sawal Is Topic Se

1. **Agar do variables ka Covariance Zero ($\text{Cov}(X, Y) = 0$) ho, toh kya wo 100% Independent hain?** -> Zaroori nahi! Zero covariance sirf **Linear Relationship** na hona prove karta hai; non-linear relationship (e.g. $Y = X^2$ where $X \sim \mathcal{U}(-1, 1)$) tab bhi exist kar sakti hai.
2. **Linearity of Expectation formula kya hota hai?** -> $\mathbb{E}[aX + bY] = a\mathbb{E}[X] + b\mathbb{E}[Y]$ (Variables independent na hon tab bhi ye hamesha TRUE rehta hai!).
3. **Variance of Sum of correlated variables kya hota hai?** -> $\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X, Y)$.

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**Next Topic:** 13. Law of Large Numbers & Central Limit Theorem (CLT) with Monte Carlo Simulations

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