Finding expectation of a random variable

Question

Let $X$ be a random variable whose distribution function is given by $$ \mathrm{F}\left(x\right) = \left\{\begin{array}{lcl} {\displaystyle 0} & \mbox{if} & {\displaystyle x < 2} \\[1mm] {\displaystyle{1 \over 3}\,x} & \mbox{if} & {\displaystyle 2 \leq x \leq 3} \\[1mm] {\displaystyle 1} & \mbox{if} & {\displaystyle x > 3} \end{array}\right. $$ Then find $E\left(X\right)$ and $E\left(X^{2}\right)$.

Given random variable is neither discrete nor continuous. Then by Jordan decomposition theorem first we have to write $X$ as a sum of a step function and a continuous function. But how should I do that ?.

Answer 1

You can use Stieltjes integral formula.

$E(X^k)=\int_R x^kdF(x)=\frac{2}{3}2^k+\int_2^3 \frac{x^k}{3}dx$