Problem

Akia selects at random a value from the continuous random variable $W$, which is uniformly distributed over the interval $[a, b]$

The probability that Akia selects a value less than $k$ is $\dfrac{1}{5}$ The probability that Akia selects a value greater than 17 is $\dfrac{53}{60}$

(a) Find the probability that Akia selects a value between 17 and $k$

It is known that $\operatorname{Var}(W) = 75$

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

(ii) Find the value of $k$

(c) Find $P(-5 < W < 5)$

(d) Find $E(W^2)$

(Total for Question 6 is 12 marks)