CAIE A-Level Mathematics 6.3.2 Probability Density Functions

Practise integrating a continuous density over its support to find constants and probabilities, then locating percentiles or calculating expectation and variance from weighted…

Exam points

set the integral of f(x) over its full support equal to one to determine an unknown constant
integrate f(x) between stated bounds and use symmetry or complements for the required area
solve an area equation for a median or percentile and use x f(x) or x²f(x) for moments

[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 ]

$\int_{0}^{m}\left(2 x-x^{3}\right) d x=$

M1

Attempt integrate f(x) with limits 0 to m (or m to $\sqrt{2}$ ) and equate to $\frac{1}{2}$ .

$m^{2}-\frac{m^{4}}{4}=\frac{1}{2}$

A1

For correct quartic in any form.

$m^{4}-4 m^{2}+2=0 \Rightarrow m^{2}=\frac{4 \pm \sqrt{16-8}}{2} \quad[=2 \pm \sqrt{2}]$

M1

For solving their three term quartic to find $m^{2}$ .

$m=\sqrt{2-\sqrt{2}}$ or 0.765(3 sf)

A1

(b)

Find the exact value of E(X).

[ 3 ]

$\int_{0}^{\sqrt{2}}\left(2 x^{2}-x^{4}\right) \mathrm{d} x$

M1

Attempt to integrate x f(x). Ignore limits.

$\left[\frac{2 x^{3}}{3}-\frac{x^{5}}{5}\right]_{0}^{\sqrt{2}}$

A1

Correct integration and correct limits.

$\left[=\frac{4 \sqrt{2}}{3}-\frac{4 \sqrt{2}}{5}\right]=\frac{8}{15} \sqrt{2}$

A1

OE For single exact term.

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6.3.2—Density functions question 1

Question (a)

Question (b)