Problem

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

$F(x) = \begin{cases} 0 & x < 0 \\ k(6x^2 - x^4) & 0 \leqslant x \leqslant 1 \\ 1 & x > 1 \end{cases}$

where $k$ is a positive constant.

(a) Show that $k = \dfrac{1}{5}$

(b) Find $P(X > 0.6)$

(c) (i) Show that the median, $m$, of $X$ satisfies

$am^4 + bm^2 + c = 0$

where $a$, $b$ and $c$ are integers to be found.

(ii) Hence, find the value of $m$

(d) Find, using calculus, the mode of $X$

Given that $E(X) = 0.64$

(e) state, giving a reason, the skewness of the distribution.

(Total for Question 1 is 11 marks)