Question

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

Answer 1

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}`