Law of Large Numbers

Theorem

Bernoulli's Theorem

Let the probability of the occurrence of an event be $p$.

Let $n$ independent trials be made, with $k$ successes.

Then:

$\ds \lim_{n \mathop \to \infty} \frac k n = p$

Weak Law of Large Numbers

Let $P$ be a population.

Let $P$ have mean $\mu$ and finite variance.

Let $\sequence {X_n}_{n \mathop \ge 1}$ be a sequence of random variables forming a random sample from $P$.

Let:

$\ds {\overline X}_n = \frac 1 n \sum_{i \mathop = 1}^n X_i$

Then:

${\overline X}_n \xrightarrow p \mu$

where $\xrightarrow p$ denotes convergence in probability.

Strong Law of Large Numbers

Let $P$ be a population.

Let $P$ have mean $\mu$ and finite variance.

Let $\sequence {X_n}_{n \mathop \ge 1}$ be a sequence of random variables forming a random sample from $P$.

Let:

$\ds {\overline X}_n = \frac 1 n \sum_{i \mathop = 1}^n X_i$

Then:

$\ds {\overline X}_n \xrightarrow {\text {a.s.} } \mu$

where $\xrightarrow {\text {a.s.} }$ denotes almost sure convergence.