Its variance, \(\sigma^2\) or \(\mathrm{Var}(X)\) is given by
\[\begin{aligned} 
\sigma^2 
&= \mathrm{Var}(X) \\ 
&= \mathrm{E}(X - \mu)^2 \\ 
&= \mathrm{E}(X^2) - (\mathrm{E}(X))^2 \\ 
&= \int_{\text{all}\hspace{0.5em} x} x^2 \mathrm{f}(x) \, \mathrm{d}x - \left[ \int_{\text{all}\hspace{0.5em} x} x \mathrm{f}(x) \, \mathrm{d}x \right]^2 
\end{aligned}
\]
Note: The standard deviation of \(X\) is \(\sigma = \sqrt{\mathrm{Var}(X)}\).

Its variance, \(\sigma^2\) or \(\mathrm{Var}(X)\) is given by

\[\begin{aligned} 
\sigma^2 
&= \mathrm{Var}(X) \\ 
&= \mathrm{E}(X - \mu)^2 \\ 
&= \mathrm{E}(X^2) - (\mathrm{E}(X))^2 \\ 
&= \int_{\text{all } x} x^2 \mathrm{f}(x) \, \mathrm{d}x - \left[ \int_{\text{all } x} x \mathrm{f}(x) \, \mathrm{d}x \right]^2 
\end{aligned}
\]

**Note:** The standard deviation of \(X\) is \(\sigma = \sqrt{\mathrm{Var}(X)}\).

Expectation and Variance of Continuous Random Variables

Continuous Random Variables

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Variance of a CRV

Variance of a CRV