What is the mathematical formula for calculating the variance of a discrete random variable?

1 Answer
Bill K.
Oct 24, 2015

Let mu_{X}=E[X]=sum_{i=1}^{infty}x_{i} * p_{i} be the mean (expected value) of a discrete random variable X that can take on values x_{1},x_{2},x_{3},... with probabilities P(X=x_{i})=p_{i} (these lists may be finite or infinite and the sum may be finite or infinite). The variance is sigma_{X}^{2}=E[(X-mu_{X})^2]=sum_{i=1}^{infty}(x_{i}-mu_{X})^2 * p_{i}

Explanation:

The previous paragraph is the definition of the variance sigma_{X}^{2}. The following bit of algebra, using the linearity of the expected value operator E, shows an alternative formula for it, which is often easier to use.

sigma_{X}^{2}=E[(X-mu_{X})^2]=E[X^2-2mu_{X}X+mu_{X}^{2}]

=E[X^2]-2mu_{X}E[X]+mu_{X}^{2}=E[X^2]-2mu_{X}^{2}+mu_{X}^{2}

=E[X^2]-mu_{X}^{2}=E[X^{2}]-(E[X])^2,

where E[X^{2}]=sum_{i=1}^{infty}x_{i}^{2} * p_{i}

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