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)=n∑i=1(xi−μ)2f(xi)Var(X)=n∑i=1(xi−μ)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(x−μ)2f(x)dxVar(X)=∫ba(x−μ)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 .