The variance of a random variable

Summary

Discrete random variable

  • If XX is a discrete random variable with range x1,x2,,xnx1,x2,,xn , probability mass function f(x)f(x) and expected value μμ then the variance of XX is defined as

Var(X)=ni=1(xiμ)2f(xi)Var(X)=ni=1(xiμ)2f(xi)

Continuous random variable

  • If XX is a continuous random variable with range [a,b][a,b] , probability density function f(x)f(x) and expected value μμ then the variance of XX is defined as

Var(X)=ba(xμ)2f(x)dxVar(X)=ba(xμ)2f(x)dx

About the variance

  • Var(X)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 nn observations).
  • The standard deviation of a random variable is defined as

SD(X)=Var(X)SD(X)=Var(X)

  • A common symbol for Var(X)Var(X) is σ2σ2 and a common symbol for SD(X)SD(X) is σσ .