1. 1
   In a game, a player spins a spinner once. The score $X$ has probability distribution $P(X = 0) = 0.1$, $P(X = 1) = 0.3$, $P(X = 2) = 0.4$ and $P(X = 3) = 0.2$. It is given that $E(X) = 1.7$.

   (a)

   Find $E(X^2)$.

   [1 mark]

   • A$2.89$
   • B$14$
   • C$3.7$
   • D$3.4$

   (b)

   Find $\mathrm{Var}(X)$.

   [1 mark]

   • A$0.81$
   • B$2.0$
   • C$0.9$
   • D$6.59$

   (c)

   To play, the player pays 8 dollars and then receives 5 dollars for every point scored. Let $Y$ be the player's gain in dollars. Find $E(Y)$ and determine whether the game is fair.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The continuous random variable $X$ has probability density function $f(x) = kx^2(3 - x)$ for $0 \le x \le 3$, and $f(x) = 0$ otherwise, where $k$ is a positive constant.

   (a)

   Find the value of $k$.

   [1 mark]

   • A$\frac{27}{4}$
   • B$\frac{1}{27}$
   • C$\frac{1}{6}$
   • D$\frac{4}{27}$

   (b)

   Find the mode of $X$.

   [1 mark]

   • A$3$
   • B$2$
   • C$1.5$
   • D$1.8$

   (c)

   Find $E(X)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The time $X$ minutes that a passenger waits for a bus is modelled by the probability density function $$f(x) = \begin{cases} kx, & 0 \le x \le 2 \\ \frac{k(6 - x)}{2}, & 2 < x \le 6 \\ 0, & \text{otherwise} \end{cases}$$ where $k$ is a positive constant.

   (a)

   Show that $k = \frac{1}{6}$.

   [3 marks]

   (b)

   Find the median waiting time, giving your answer in exact form and correct to three significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A car park has a maximum stay of 4 hours. The time $T$ hours that a car stays is modelled by the probability density function $f(t) = \frac{t}{8}$ for $0 \le t \le 4$, and $f(t) = 0$ otherwise. The charge for a stay of $T$ hours is $C$ dollars, where $C = 3 + 2.5T$.

   (a)

   Find $E(T)$ and $\mathrm{Var}(T)$.

   [6 marks]

   (b)

   (i) Find the mean and the standard deviation of the charge $C$.
   (ii) Find the probability that a randomly chosen car pays more than the mean charge.

   [6 marks]

   Total for question 4: 12 marks

End of questions