13. Law of Large Numbers & Central Limit Theorem (CLT) with Monte Carlo Simulations • Mathematics, Statistics & Machine Learning (Big Tech Prep) • Rishabh Ki Kaksha

Mathematics, Statistics & Machine Learning (Big Tech Prep) › 13. Law of Large Numbers & Central Limit Theorem (CLT) with Monte Carlo Simulations

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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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# 13. Law of Large Numbers & Central Limit Theorem (CLT) with Monte Carlo Simulations

> Mathematics & ML Foundations Notes by Rishabh - Topic 13

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### 1. Law of Large Numbers (LLN) 🪙

Sample size $n \rightarrow \infty$ grow hone par Sample Mean $\bar{X}_n$ True Population Mean $\mu$ par mathematically **converge** ho jati hai:
$$P\left( \lim_{n \to \infty} \bar{X}_n = \mu \right) = 1$$

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### 2. Central Limit Theorem (The Miracle of Statistics 🪄)

Chahe population ka original distribution kaisa bhi ho (Uniform, Exponential, Skewed, Bimodal):  
**Sample Means ka distribution hamesha Gaussian (Normal Distribution) ban jata hai!**
$$\bar{X}_n \sim \mathcal{N}\left( \mu, \frac{\sigma^2}{n} \right) \quad \text{as } n \ge 30$$

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### 3. Monte Carlo Simulation for Pi ($\pi$) Estimation

```python
import numpy as np

N = 10_000_000
x = np.random.uniform(-1.0, 1.0, N)
y = np.random.uniform(-1.0, 1.0, N)

# Distance from origin inside unit circle:
inside_circle = (x**2 + y**2) <= 1.0
estimated_pi = 4.0 * np.sum(inside_circle) / N
print(f"Estimated Pi: {estimated_pi:.5f} (True: 3.14159)")
```

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

1. **Standard Error of the Mean (SEM) sample size ke sath kaise scale karta hai?** -> $\text{SEM} = \frac{\sigma}{\sqrt{n}}$ (Sample size $4\times$ badhane par error sirf $2\times$ kam hota hai).
2. **Central Limit Theorem kab fail hoti hai?** -> Jab underlying distribution ka Variance infinite ho (e.g. Cauchy Distribution ya extreme Pareto Power Law tails).
3. **Monte Carlo Tree Search (MCTS) AlphaGo aur DeepSeek reasoning me kaise use hota hai?** -> Random rollouts aur simulation sampling se uncomputable game tree trajectories ki expected value estimate karna.

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**Next Topic:** 14. Hypothesis Testing: Null Hypothesis, p-Values, Z-Test, Student's t-Test & Chi-Square Test

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