Probability MT22, Weak law of large numbers
Flashcards
Let $X _ 1, \ldots, X _ n$ be a random sample (i.i.d. r.v.s.) with mean $\mu$. What does the weak law of large numbers state for any $\epsilon > 0$ (as $n \to \infty$)?
\[\lim _ {n \to \infty}\mathbb{P}\left(\left \vert \frac{1}{n} \sum^n _ {i=1} X _ i - \mu\right \vert > \epsilon\right) = 0\]
The weak law of large numbers 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 (under the assumption variance is finite) the weak law of large numbers:
Suppose that $X _ 1, \ldots$ are i.i.d. random variables with mean $\mu$. Then, for all $\varepsilon > 0$,
\[\mathbb P \left( \left \vert \frac 1 n \sum^n _ {i=1} X _ i - \mu \right \vert \le \varepsilon \right) \to 0\]
as $n \to \infty$.
Suppose that $X _ 1, \ldots$ are i.i.d. random variables with mean $\mu$. Then, for all $\varepsilon > 0$,
@todo (probability, page 65).
@important~