Mathematics 9709 · AS & A Level · Continuous random variables
Continuous random variables — practice question
The probability density function for the random variable $X$ is defined by $f(x) = \begin{cases} kx(4 - x) & 0 \le x \le 2, \\ 0 & \text{otherwise}, \end{cases}$ where $k$ is a constant.
(a(i))[3]
Show, using your result, that $k = \frac{3}{16}$.
(a(ii))[3]
Find the value of $\mathrm{E}(X)$.
(b)[3]
The random variable $Y$ has the following features:
• $Y$ can take values only from $0$ to $5$.
• The probability density function of $Y$ is symmetrical.
Given that $\mathrm{P}(Y < a) = 0.2$, find $\mathrm{P}(2.5 < Y < 5 - a)$ and illustrate your method with a sketch on the axes provided.
Worked solution & mark scheme
This 9-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Sets up the equation $k\int_0^2 (4x-x^2)\,dx=1$” …