Problem

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

$$F(x)= \begin{cases} 0, & x < 0,\\ 1-\frac{1}{144}(12-x)^2, & 0 \le x \le 12,\\ 1, & x > 12. \end{cases}$$

(a) Find the upper quartile of $X$.

(b) Find $\operatorname{Var}(X^2)$.

The random variable $Y$ is given by $Y=\sqrt{X}$.

(c) Find the probability density function of $Y$.