Problem

The random variable $X$ has cumulative distribution function given by

$$F(x)= \begin{cases} 0, & x<0 \\ \dfrac{1}{100}(ax^3+bx^2+15x), & 0 \leq x \leq 5 \\ 1, & x>5 \end{cases}$$

Given that $E(X^2)=6.25$

(a) show that $6a+b=0$

(b) find the value of $a$ and the value of $b$

(c) find P($3 \leq X \leq 7$)