Question

Let X be a continuous r.v with density function f(x) =(|x| 10 , for −2 < x < 4 0, elsewhere
. Calculate the expected value of X.

Answer 1

Let $X$ be a continuous r.v with density function

$$f(x) = \begin{cases} {|x|\over 10}, &\text{for}\ -2<x<4 \\ 0 &\text{elsewhere} \end{cases}$$

Calculate the expected value of $X$

$$E(X)=\mu_X=\displaystyle\int_{-\infin}^{\infin}xf(x)dx=$$

$$=\displaystyle\int_{-2}^{0}x({-x\over 10})dx+\displaystyle\int_{0}^{4}x({x\over 10})dx=$$

$$=\bigg[-{x^3\over 30}\bigg]\begin{matrix} 0 \\ -2 \end{matrix}+\bigg[{x^3\over 30}\bigg]\begin{matrix} 4 \\ 0 \end{matrix}={8\over 30}+{64\over 30}=2.4$$

$E(X)=2.4$