Random sample
Definition taught in Statistical inference, part of Statistical inference
Random sample
| Type | Definition |
| Taught in | Statistical inference |
| Part | Statistical inference |
| Properties | 2 |
| Questions using it | 7 |
X_1,\ldots,X_n are a random sample from a distribution F when they are independent and each has distribution F. The joint density or mass function is then the product of the marginals.
Properties
Moments of the sample mean and variance
Main article: Moments of the sample mean and variance
If X_1,\ldots,X_n are a random sample from a distribution with mean \mu and variance \sigma^2, then
\mathbb{E}\overline X=\mu,\qquad \operatorname{Var}(\overline X)=\frac{\sigma^2}{n},\qquad \mathbb{E} s^2=\sigma^2.
No normality is needed.
Normal sample theorem
Main article: Normal sample theorem
If X_1,\ldots,X_n are a random sample from \operatorname{N}(\mu,\sigma^2), then
- \overline X is \operatorname{N}(\mu,\sigma^2/n),
- (n-1)s^2/\sigma^2 is \chi^2_{n-1}, and
- \overline X and s^2 are independent.
Used in
- Taught in Statistical inference, where the derivation and examples are.
- Also used in Joint distributions · Estimation · Confidence intervals · Hypothesis tests · Linear regression · Studies, data, and evidence · Estimation examples
- Questions that use it (7) — Which of these are statistics? · The expectation of the sample variance · Bias of the two variance estimators · An exponential rate · A Poisson mean · A shrunken sample mean · Tyre wear on five cars (answer)
- All 4 questions on Statistical inference
See also
Joint probability density · Discrete distribution · Margin of error · Maximum likelihood · Moments of the sample mean and variance · Normal sample theorem · One-sample t test · Simple random sample