Problem

The function $f(x)$ is defined as

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

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

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

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.