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

Question

Hunter Aldridge · Accepted Answer

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}# 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}#

Jonathan Allen · Answer

undefined The mathematical formula for calculating the variance of a discrete random variable $X$ is:

$$Var(X) = \sum_{i=1}^{n} (x_i - \mu)^2 \cdot P(X = x_i)$$

Where:
- $Var(X)$ represents the variance of the random variable $X$.
- $x_i$ are the possible values of the random variable.
- $\mu$ is the mean (expected value) of the random variable, given by $\mu = \sum_{i=1}^{n} x_i \cdot P(X = x_i)$.
- $P(X = x_i)$ is the probability mass function, representing the probability that the random variable takes the value $x_i$.
- $n$ is the total number of distinct values that the random variable can take.

This formula computes the average of the squared differences between each value of the random variable and its mean, weighted by the probabilities of those values occurring. It provides a measure of how spread out the values of the random variable are from its mean.