Mathematics 9709 · AS & A Level · Continuous random variables
Continuous random variables — practice question
The diagram displays the graph of the probability density function, $f$, for a random variable $X$ that can take only values from $x = 0$ to $x = 3$. The graph is symmetric about the line $x = 1.5$.
(a)[2]
You are told that $P(X < 0.6) = a$ and $P(0.6 < X < 1.2) = b$. Determine $P(0.6 < X < 1.8)$ in terms of $a$ and $b$.
(b(i))[3]
It is now stated that the probability density function of $X$ is $$f(x) = \begin{cases} kx^2(3 - x)^2, & 0 \le x \le 3, \\ 0, & \text{otherwise}, \end{cases}$$ where $k$ is a constant. Show that $k = \frac{10}{81}$.
(b(ii))[3]
Find the value of $\mathrm{Var}(X)$.
Worked solution & mark scheme
This 8-mark question has a full step-by-step worked solution and mark scheme. One marking point: “A correct expression, simplified or not, for the probability, for instance $1-2a-b$” …