Law of Large Numbers Demo | Sample Mean Convergence

📖 Tutorial | Law of Large Numbers

1. Definition

Law of Large Numbers (LLN): for i.i.d. X₁,…,Xₙ with mean μ, the sample mean X̄ₙ converges in probability to μ as n grows. The more trials, the more stable the average.

2. Symbols

Symbol	Meaning
X̄ₙ	mean of n samples
μ=E(X)	population expectation
n	sample size
convergence in prob.	larger n ⇒ smaller chance of a large deviation

3. How it works

A fixed seed gives reproducible samples;
It computes the mean for n=10,100,1000,…;
The mean should hug the theory as n grows;
Try uniform, Bernoulli or exponential.

4. Steps

Pick a distribution;
Enter a comma-separated list of increasing n;
Click Calculate and read each row;
Check whether the largest n is closest to μ.

5. Example

Example: Uniform U(0,1), mean 0.5. At n=10 the mean may be 0.4–0.6; at n=100000 it usually lands near 0.495–0.505, close to 0.5.

6. Pitfalls

Large n only guarantees closeness “with high probability”, not every run;
A fixed seed is reproducible but the trend is the same with another seed;
The sample mean is random—do not treat one run as a proof.

❓ FAQ | Law of Large Numbers

Why does the mean approach the expectation?

Fluctuations cancel over many independent repetitions, leaving the stable expected term.

Does larger n always mean closer?

Convergence is in probability: very likely closer, but occasional deviation remains.

What are the three expectations?

Uniform U(0,1): 0.5; Bernoulli p=0.5: 0.5; Exponential λ=1: 1.

How is it different from the CLT?

LLN: the mean converges to μ; CLT: the fluctuations of the mean are normal.

Why use a fixed seed?

To make results reproducible for teaching; real Monte Carlo differs but converges the same.