Mean and Variance of a Random Variable

Mean (Expectation) of a Random Variable

The mean, or expectation, of a discrete random variable $X$ with probability distribution $P(X=x_i)=p_i$ is defined as

The mean is the probability-weighted average of the values $X$ can take -- it represents the long-run average value of $X$ over a very large number of repetitions of the experiment, and need not itself be a value $X$ can actually take.

Variance and Standard Deviation

The variance of $X$ measures how spread out the distribution is about the mean, and is defined as the expectation of the squared deviation from the mean:

Exactly as with the variance of a data set, this expands algebraically into the far more convenient computational shortcut

The standard deviation of $X$ is defined as the (positive) square root of the variance, $\sigma_X=\sqrt{\operatorname{Var}(X)}$, and is expressed in the same units as $X$ itself (whereas the variance is in squared units).

Worked Illustration

For $X=$ number of tails in three coin tosses (distribution $\tfrac18,\tfrac38,\tfrac38,\tfrac18$ for $x=0,1,2,3$, Section 6):

$\operatorname{Var}(X) = E(X^2)-[E(X)]^2 = 3-\left(\frac32\right)^2 = 3-\frac94=\frac34.$ …