Problem

The function $f(x)$ is defined as

$f(x) = \begin{cases} \dfrac{1}{9}(x-5)(3-x) & 1 \leqslant x \leqslant 4 \\ 0 & \text{otherwise} \end{cases}$

Albert believes that $f(x)$ is a valid probability density function.

(a) Sketch $f(x)$ and comment on Albert’s belief.

The continuous random variable $Y$ has probability density function given by

$g(y) = \begin{cases} ky(12 - y^2) & 1 \leqslant y \leqslant 3 \\ 0 & \text{otherwise} \end{cases}$

where $k$ is a positive constant.

(b) Use calculus to find the mode of $Y$

(c) Use algebraic integration to find the value of $k$

(d) Find the median of $Y$ giving your answer to 3 significant figures.

(e) Describe the skewness of the distribution of $Y$ giving a reason for your answer.

(Total for Question 4 is 14 marks)