11. Probability Distributions: Gaussian (Normal), Bernoulli, Binomial, Poisson & Exponential Distributions • Mathematics, Statistics & Machine Learning (Big Tech Prep) • Rishabh Ki Kaksha

Mathematics, Statistics & Machine Learning (Big Tech Prep) › 11. Probability Distributions: Gaussian (Normal), Bernoulli, Binomial, Poisson & Exponential Distributions

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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

Interactive Lesson Studio

# 11. Probability Distributions: Gaussian (Normal), Bernoulli, Binomial, Poisson & Exponential Distributions

> Mathematics & ML Foundations Notes by Rishabh - Topic 11

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### 1. The Core Probability Distributions in AI / ML 📊

1. **Bernoulli Distribution ($p$):** Single coin toss (Binary 0 or 1).
2. **Binomial Distribution ($n, p$):** $n$ independent coin tosses me $k$ successes ka count.
3. **Gaussian (Normal) Distribution ($\mu, \sigma^2$ - The Bell Curve 🔔):**
   $$f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left( -\frac{(x - \mu)^2}{2\sigma^2} \right)$$
4. **Poisson Distribution ($\lambda$):** Fixed time interval me rare events occur hone ka count (e.g. Server requests per second).

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### 2. Master Comparison Matrix

| Distribution | Discrete / Continuous | Parameters | Mean | Variance |
| :--- | :--- | :--- | :--- | :--- |
| **Bernoulli** | Discrete | $p$ | $p$ | $p(1-p)$ |
| **Binomial** | Discrete | $n, p$ | $np$ | $np(1-p)$ |
| **Gaussian (Normal)**| Continuous | $\mu, \sigma^2$ | $\mu$ | $\sigma^2$ |
| **Poisson** | Discrete | $\lambda$ | $\lambda$ | $\lambda$ |
| **Exponential**| Continuous | $\lambda$ | $\frac{1}{\lambda}$ | $\frac{1}{\lambda^2}$ |

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

1. **Multivariate Gaussian Distribution me Covariance Matrix $\Sigma$ ka shape kya batata hai?** -> Diagonal elements individual variable variances batate hain; off-diagonal elements variables ke beech ka linear relationship/correlation dictate karte hain (Elliptical contour orientations).
2. **68-95-99.7 Empirical Rule in Normal Distribution kya hoti hai?** -> 68.2% data $\mu \pm 1\sigma$ me, 95.4% data $\mu \pm 2\sigma$ me, aur 99.7% data $\mu \pm 3\sigma$ me lie karta hai.
3. **VAE (Variational Autoencoders) me Gaussian Prior $\mathcal{N}(0, I)$ kyu use hota hai?** -> Latent space ko smooth aur continuous regularize karne ke liye using KL Divergence.

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**Next Topic:** 12. Expectation, Variance, Covariance & Pearson Correlation Coefficient

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