Variance

Study guides: Exam VEE-MS, Exam P-1, Exam FAM, Exam SRM, Exam PA, Exam ASTAM, Exam MAS-I, Exam MAS-II, Exam 8

Variance $\sigma^2$ measures the spread or dispersion of a Random Variable's distribution as the expected squared deviation from the mean.

Efficient computation formula: $\text{Var}(X) = E[X^2] - \mu^2$ where $\mu = E[X]$
$\text{Var}(aX + b) = a^2 \text{Var}(X)$ — scaling changes variance, shifts do not
$\text{Var}(X) \geq 0$ , since $(X - \mu)^2 \geq 0$ ; a variable that always equals its mean, such as a constant, has variance 0

$\text{Var}(X) = E\left[(X - \mu)^2\right]$

$= E[X^2] - \mu^2$

Variance of an Insurance Payment A loss $X$ has $E[X] = 500$ and $E[X^2] = 310{,}000$ . Find $\text{Var}(X)$ .

Answer Using the computational formula: $\text{Var}(X) = E[X^2] - (E[X])^2 = 310{,}000 - 500^2 = 310{,}000 - 250{,}000 = 60{,}000$