For a discrete random variable X with mean mu=E(X), variance is the probability-weighted mean of the squared deviations from mu. Assume the second moment is finite. Var(X)=E[(X-mu)2]= _x(x-mu)2P(X=x). Expanding the square gives a useful calculation formula: Var(X)=E(X2)-2 (X)+mu2 =E(X2)-mu2. Calculate E(X2)= _xx2P(X=x), then subtract [E(X)]2. In each sum, include every possible value of X. Variance is non-negative. The standard deviation is sigma=sqrt(Var(X)), in the same units as X.

For a discrete random variable \(X\) with mean \(\mu=\mathrm{E}(X)\), variance is the probability-weighted mean of the squared deviations from \(\mu\). Assume the second moment is finite.

\(\operatorname{Var}(X)=\mathrm{E}[(X-\mu)^2]=\sum\limits_{x}(x-\mu)^2P(X=x)\).

Expanding the square gives a useful calculation formula:

\(\begin{aligned}\operatorname{Var}(X)&=\mathrm{E}(X^2)-2\mu\mathrm{E}(X)+\mu^2\\&=\mathrm{E}(X^2)-\mu^2.\end{aligned}\)

Calculate \(\mathrm{E}(X^2)=\sum\limits_{x}x^2P(X=x)\), then subtract \([\mathrm{E}(X)]^2\). In each sum, include every possible value of \(X\).

Variance is non-negative. The standard deviation is \(\sigma=\sqrt{\operatorname{Var}(X)}\), in the same units as \(X\).

Discrete Random Variables (DRV)

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Variance of a Discrete Random Variable

Variance of a Discrete Random Variable