Mathematics 9709 · AS & A Level · Continuous random variables
Continuous random variables — practice question
A ball is released down a slope in a game and then travels along a track until it comes to rest. Let the distance, in metres, covered by the ball be represented by the random variable $X$ with probability density function
$f(x) = \begin{cases} -k(x - 1)(x - 3), & 1 \le x \le 3, \\ 0, & \text{otherwise}. \end{cases}$
where $k$ is a constant.
(a)[1]
Explain, without doing any calculation, why $\text{E}(X) = 2$.
(b)[3]
Show that $k = \frac{3}{4}$.
(c)[3]
Find $\operatorname{Var}(X)$.
(d)[4]
One turn involves rolling the ball 3 times and recording the greatest value of $X$ obtained. If this greatest value exceeds $2.5$, the player scores a point. Find the probability that the player scores a point on a particular turn.
Worked solution & mark scheme
This 11-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Recognising it as a quadratic graph and therefore symmetric” …