CAIE A-Level Mathematics A2 6.3.2 Density Functions Questions

Practise finding probabilities, means, medians and percentiles from continuous distributions.

Exam points

calculate probabilities from a continuous probability density function
find means, medians and percentiles using integration
determine unknown parameters from continuous distribution conditions

[Maximum number: 7]

A random variable X has probability density function f given by

f(x)= \begin{cases}a x-x^{3} & 0 \leqslant x \leqslant \sqrt{2} \\ 0 & \text { otherwise }\end{cases}

where a is a constant.

(a)

Find the median of X.

[ 4 ]

$\int_{0}^{m}\left(2 x-x^{3}\right) d x=\frac{1}{2}$ $m^{2}-\frac{m^{4}}{4}=\frac{1}{2}$ $m^{4}-4 m^{2}+2=0 \Rightarrow m^{2}=\frac{4 \pm \sqrt{16-8}}{2} \quad[=2 \pm \sqrt{2}]$ $m=\sqrt{2-\sqrt{2}}$ or 0.765 (3 sf)

(b)

Find the exact value of E(X).

[ 3 ]

$\int_{0}^{\sqrt{2}}\left(2 x^{2}-x^{4}\right) \mathrm{d} x$ $\left[\frac{2 x^{3}}{3}-\frac{x^{5}}{5}\right]_{0}^{\sqrt{2}}$ $\left[=\frac{4 \sqrt{2}}{3}-\frac{4 \sqrt{2}}{5}\right]=\frac{8}{15} \sqrt{2}$

All question bank results loaded

CAIE A-Level Mathematics A2 6.3.2 Density Functions Questions question 1

Question (a)

Question (b)