Probability MT22, Weak law of large numbers
Flashcards
@State the weak law of large numbers for a random sample (i.i.d. r.v.s.) $X _ 1, \ldots, X _ n$ with mean $\mu$, for any $\epsilon > 0$.
\[\lim _ {n \to \infty}\mathbb{P}\left(\left \vert \frac{1}{n} \sum^n _ {i=1} X _ i - \mu\right \vert > \epsilon\right) = 0\]
The weak law of large numbers (∆weak-law-of-large-numbers) states that
\[\lim _ {n \to \infty}\mathbb{P}\left(\left \vert \frac{1}{n} \sum^n _ {i=1} X _ i - \mu\right \vert > \epsilon\right) = 0\]
What is the interpretation of what this means?
\[\overline{X _ n} \approx \mu \text{ for large } n.\]
@Prove the weak law of large numbers, under the assumption that the variance is finite (∆weak-law-of-large-numbers): suppose that $X _ 1, \ldots$ are i.i.d. random variables with mean $\mu$. Then, for all $\varepsilon > 0$,
\[\mathbb P \left( \left \vert \frac 1 n \sum^n _ {i=1} X _ i - \mu \right \vert > \varepsilon \right) \to 0\]
as $n \to \infty$.
@todo (probability, page 65).