Answer
$\mu=5$
$Var(X)=\frac{25}{3}$
$\sigma=\frac{5\sqrt 3}{3}$
Work Step by Step
We are given $f(x)=\frac{1}{10}; [0;10]$
The expected value $\mu=\int^{10}_{0}xf(x)dx$ $=\int^{10}_{0}x(\frac{1}{10})dx$
$=\frac{1}{10}(\frac{x^{2}}{2})|^{10}_{0}$
$=\frac{1}{10}(50-0)=5$
The variance is
$Var(X)=\int^{10}_{0}(x-5)^{2}f(x)dx$
$=\int^{10}_{0}(x^{2}-10x+25)\frac{1}{10}dx$
$=\frac{1}{10}\int^{10}_{0}(x^{2}-10x+25)dx$
$=\frac{1}{10}(\frac{x^{3}}{3}-5x^{2}+25x)|^{10}_{0}$
$=\frac{25}{3}$
The standard deviation of X is
$\sigma=\sqrt Var(X)$
$=\sqrt \frac{25}{3}=\frac{5\sqrt 3}{3}$
