Question

0 x/8 x <0 0 11, Find the mean when F(x) = x < 2

Answer 1

The distribution function of random variable X is

$F(x) =\begin{matrix} 0 & x< 0\\ \frac{x}{8} & 0\leq x< 2\\ \frac{x^{2}}{16} & 2\leq x< 4 \end{matrix}$

The probability distribution function of X is

$f(x) = F'(x) = \frac{d}{dx}(F(x))$

$f(x) =\begin{matrix} 0 & x< 0\\ \frac{1}{8} & 0\leq x< 2\\ \frac{x}{8} & 2\leq x< 4 \end{matrix}$

$Mean= E(X) = \int x*f(x) dx$

$E(X) = \frac{1}{8}\int_{0}^{2}x*dx &plus;\frac{1}{8}\int_{2}^{4}x^{2}*dx$

$E(X) = \frac{1}{8}\left [ \frac{x^{2}}{2} \right ]_{0}^{2} &plus;\frac{1}{8}\left [ \frac{x^{3}}{3} \right ]_{2}^{4}$

$E(X) = \frac{1}{4} &plus; \frac{56}{24} = \frac{62}{24}$

E(X) = 2.5833