Problem

A continuous random variable $X$ has cumulative distribution function $F$ given by

$$F(x)= \begin{cases} 0, & x < 1,\\ \frac{1}{5}x+a, & 1 \le x < 4,\\ \frac{1}{50}x^2+b, & 4 \le x \le 6,\\ 1, & x > 6, \end{cases}$$

where $a$ and $b$ are constants.

(a) Find the value of $a$ and the value of $b$.

(b) Find the probability density function of $X$.

(c) Given that $\mathrm{E}(X)=\frac{529}{150}$, find $\operatorname{Var}(X)$.

(d) Find the 10th and 90th percentiles of $X$.