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2.1.5 Mean, Variance and Standard deviation

Two important characteristics of random variables are their mean ``average'' and standard deviation ``spread'' or ``scale''

Definition 8   If $ X$ is a discrete r.v. with distribution function $ P_X(x)$ then the mean of $ X$ is:

$\displaystyle \mu_X = \sum_x x P_X(x) $

and the variance of $ X$ is:

$\displaystyle \sigma^2_X = \sum_x (x-\mu_X)^2 P_X(x) $

The standard deviation $ \sigma_X$ of $ X$ is:

$\displaystyle \sigma_X = \sqrt{\sigma_X^2} $

Similarly for a continuous r.v. we have:

Definition 9   If $ X$ is a continuous r.v. with density function $ f_X(x)$ then the mean of $ X$ is:

$\displaystyle \mu_X = \int_x x f_X(x)dx $

and the variance of $ X$ is:

$\displaystyle \sigma^2_X = \int_x (x-\mu_X)^2 f_X(x)dx $

The standard deviation $ \sigma_X$ of $ X$ is:

$\displaystyle \sigma_X = \sqrt{\sigma_X^2} $

Remarks

  1. $ \mu_X$ is a weighted average
  2. $ \sigma_X^2$ is the (weighted) average squared distance from $ \mu_X$



Eran Borenstein
2007-05-04