Probability MT22, Random samples

Created: October 21, 2025 | Read markdown | About these notes | View in context | Study these flashcards |

Flashcards

What is

\[\mathrm{Var}\left( \sum^n _ {i=1} X _ i \right)\]

?

variance-of-a-sum
\[\sum _ {i=1}^n \mathrm{Var}(X _ i) + 2\sum _ {1 \le i < j \le n} \mathrm{Cov}(X _ i, X _ j)\]

@Define a random sample of a distribution.

random-sample

Independent random variables $X _ 1, X _ 2, \ldots, X _ n$ with the same distribution.

Given a random sample (i.i.d. r.v.s.) $X _ 1, \ldots, X _ n$, @define the sample mean $\overline{X _ n}$.

sample-mean
\[\frac{1}{n} \sum^n _ {i=1} X _ i\]

Given a random sample (i.i.d. r.v.s.) $X _ 1, \ldots, X _ n$ from a distribution with mean $\mu$, what is the expectation $\mathbb{E}[\overline{X _ n}]$ of the sample mean?

sample-mean-expectation
\[\mu\]

Given a random sample (i.i.d. r.v.s.) $X _ 1, \ldots, X _ n$ from a distribution with variance $\sigma^2$, what is the variance $\mathrm{Var}(\overline{X _ n})$ of the sample mean?

sample-mean-variance
\[\frac{\sigma^2}{n}\]

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 (∆sample-mean-expectation, ∆sample-mean-variance)

  1. $\mathbb E [\overline X _ n] = \mu$
  2. $\mathrm{var}(\overline X _ n) = \frac {\sigma^2} n$
sample-mean-moments-proof

@todo (probability, page 63).

@important~



Related posts