# The weak law of large numbers

## Weak law of large numbers

### Weak law of large numbers

The sample mean is:

$$\bar X_n=\dfrac{1}{n}\sum_{i=1}^nX_i$$

The variance of this is:

$$Var[\bar X_n]=Var[\dfrac{1}{n}\sum_{i=1}^nX_i]$$

$$Var[\bar X_n]=\dfrac{1}{n^2}nVar[X]$$

$$Var[\bar X_n]=\dfrac{\sigma^2}{n}$$

We know from Chebyshev’s inequality:

$$P(|X-\mu | \ge k\sigma )\le \dfrac{1}{k^2}$$

Use $$\bar X_n$$ as $$X$$:

$$P(|\bar X_n-\mu | \ge \dfrac{k\sigma }{\sqrt n})\le \dfrac{1}{k^2}$$

Update $$k$$ so $$k:=\dfrac{k\sqrt n}{\sigma}$$

$$P(|\bar X_n-\mu | \ge k)\le \dfrac{\sigma^2}{nk^2}$$

As $$n$$ increases, the chance that the sample mean lies outside a given distance from the population mean approaches $$0$$.