$\text{Var}(X) = E\!\left[(X - \mu)^2\right] = E[X^2] - \mu^2$ $\text{where } \mu = E[X]$ Variance ($\sigma^2$ or $\text{Var}(X)$) is the expected squared deviation of a random variable $X$ from its mean $\mu$, measuring the spread or dispersion of its distribution. The computational formula $\text{Var}(X) = E[X^2] - (E[X])^2$ is often more efficient. Key properties include: $\text{Var}(aX + b) = a^2\,\text{Var}(X)$ and $\text{Var}(X) \geq 0$, with equality only if $X$ is constant. > [!example]- Variance of an Insurance Payment {💡 Example} > A loss $X$ has $E[X] = 500$ and $E[X^2] = 310{,}000$. Find $\text{Var}(X)$. > > > [!answer]- Answer > > Using the computational formula: > > $\text{Var}(X) = E[X^2] - (E[X])^2 = 310{,}000 - 500^2 = 310{,}000 - 250{,}000 = 60{,}000$