Answer
$\mu = 1.3$
$\sigma^2=1.81$
$\sigma=1.345$
Work Step by Step
Use the weighted mean formula, we have
$\mu = (0.4×0+0.2×1+0.2×2+0.1×3+0.1×4) / (0.4+0.2+0.2+0.1+0.1) = 1.3$
Use the weighted variance formula, we have
$\sigma^2=( (0.4×(0-1.3)²+0.2×(1-1.3)²+0.2×(2-1.3)²+0.1×(3-1.3)²+0.1×(4-1.3)²) / (0.4+0.2+0.2+0.1+0.1) ) =1.81$
And we can find the standard deviation as $\sigma=\sqrt {\sigma^2}=1.345$
