Variance of a weighted uniform distribution

Question

Given a weighted uniform distribution, where it is a 60-40 mixture of uniform(0,7.5) and uniform(7.5,10),

I have found the mean to be $$E(X) = 0.6(7.5/2) + 0.4((10+7.5)/2)$$

How do I find the variance? Help would be greatly appreciated

Answer 1

Just apply the definitions:

$\begin{align} \mathsf {Var}(X) & = \mathsf E\big(X^2\big)-{\mathsf E(X)}^2 \\[2ex] \mathsf E(X) & = \frac{6}{10}\int_0^{7.5}\frac{1}{7.5}x\operatorname d x+\frac{4}{10}\int_{7.5}^{10}\frac{1}{2.5}x\operatorname d x \\[2ex] \mathsf E(X^2) & = \frac{6}{75}\int_0^{7.5}x^2\operatorname d x+\frac{4}{25}\int_{7.5}^{10}x^2\operatorname d x \end{align}$