The variance of a random variable

Summary

Discrete random variable

• If $X$ is a discrete random variable with range ${x_1,x_2,…,x_n}$ , probability mass function $f(x)$ and expected value $μ$ then the variance of $X$ is defined as

$$Var\left( X \right)=\sum_{i=1}^{n}{ {\left( x_i-μ \right)}^2f(x_i) }$$

Continuous random variable

• If $X$ is a continuous random variable with range $\left[ a,b \right]$ , probability density function $f(x)$ and expected value $μ$ then the variance of $X$ is defined as

$$Var\left( X \right)=\int_{a}^{b}{ {\left( x-μ \right)}^2f\left( x \right)dx }$$

About the variance

• $Var(X)$ is always greater than or equal to zero.
• Do not confuse the variance (which is a property of a random variable) with the sample variance (which is computed from a sample of $n$ observations).
• The standard deviation of a random variable is defined as

$$SD\left( X \right)=\sqrt{Var(X)}$$

• A common symbol for $Var(X)$ is $σ^2$ and a common symbol for $SD(X)$ is $σ$ .