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Probability: Difference between revisions
→Expectation and Variance
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This one is not used as often on tests as total expectation | This one is not used as often on tests as total expectation | ||
{{hidden | Proof | | {{hidden | Proof | | ||
===Sample Mean and Variance=== | |||
The sample mean is <math>\bar{X} = \frac{1}{n}\sum_{i=1}^{n}X_i</math>.<br> | |||
The unbiased sample variance is <math>S^2 = \frac{1}{n-1}\sum_{i=1}^{n}(X_i - \bar{X})^2</math>. | |||
====Student's Theorem==== | |||
Let <math>X_1,...,X_n</math> be from <math>N(\mu, \sigma^2)</math>.<br> | |||
Then the following results about the sample mean <math>\bar{X}</math> | |||
and the unbiased sample variance <math>S^2</math> hold: | |||
* <math>\bar{X}</math> and <math>S^2</math> are independent | |||
* <math>\bar{X} \sim N(\mu, \sigma^2 / n)</math> | |||
* <math>(n-1)S^2 / \sigma^2 \sim \chi^2(n-1)</math> | |||
}} | }} | ||