Probability MT22, Random samples
Flashcards
What is
\[\text{Var}\left( \sum^n _ {i=1} X _ i \right)\]
?
\[\sum _ {i=1}^n \text{Var}(X _ i) + 2\sum _ {1 \le i < j \le n} \text{Cov}(X _ i, X _ j)\]
What is a random sample of a distribution?
Independent random variables $X _ 1, X _ 2, \ldots, X _ n$ with the same distribution.
Given a random sample (i.i.d. r.v.s.) $X _ 1, \ldots, X _ n$, what is the sample mean $\overline{X _ n}$?
\[\frac{1}{n} \sum^n _ {i=1} X _ i\]
Given a random sample (i.i.d. r.v.s.) $X _ 1, \ldots, X _ n$, what is the sample mean $\mathbb{E}[\overline{X _ n}]$?
\[\mu\]
Given a random sample (i.i.d. r.v.s.) $X _ 1, \ldots, X _ n$, what is the sample mean $\text{Var}(\overline{X _ n})$?
\[\frac{\sigma^2}{n}\]
Suppose that $X _ 1, X _ 2, \ldots, X _ n$ form a random sample from a distribution with mean $\mu$ and variance $\sigma^2$. @Prove that
- $\mathbb E [\overline X _ n] = \mu$
- $\text{var}(\overline X _ n) = \frac {\sigma^2} n$
@todo (probability, page 63).
@important~