5.4.3—Expectation and variance

For discrete X, E(X)=Σxp(x), E(X²)=Σx²p(x), and Var(X)=E(X²)−[E(X)]². Standard deviation is the square root of variance.

Variance is measured in squared units; standard deviation returns to the original units. For aX+b, E=aE(X)+b and Var=a²Var(X).

If E(X)=4 and Var(X)=9, then Var(2X+1)=36 and E(2X+1)=9.

Adding a constant changes the mean but not the variance; multiplying by a changes variance by a², not a.