CAIE A-Level Mathematics A2 6.3 Continuous Random Variables Questions

Practise integrating probability density functions to find constants, probabilities, expectations, medians and percentiles.

Exam points

set the total area under a density function equal to 1 to determine constants
integrate f(x) or xf(x) over the correct support for probabilities and means
find a median or percentile by equating the cumulative probability to its target

Covered syllabus points

6.3.1—Continuous random variables 6.3.2—Density functions

[Maximum number: 10]

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)

Show that a=2.

[ 3 ]

$\int_{0}^{\sqrt{2}}\left(a x-x^{3}\right) \mathrm{d} x=1$ $\left[a \frac{x^{2}}{2}-\frac{x^{4}}{4}\right]_{0}^{\sqrt{2}}=1$ $a-\frac{4}{4}=1$ a=2

(b)

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)

(c)

Find the exact value of E(X).

[ 3 ]

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

[Maximum number: 8]

(a)

Figure for Question (a) — CAIE A-Level Mathematics A2

The graph of the function f is a straight line segment from (0,0) to (2,1).
Show that f could be a probability density function.

[ 2 ]

(b)

Figure for Question (b) — CAIE A-Level Mathematics A2

The graph of the function g is a semicircle, centre (0,0), entirely above the x-axis.
Given that g is a probability density function, find the radius of the semicircle.

[ 2 ]

(c)

Figure for Question (c) — CAIE A-Level Mathematics A2

The time, X minutes, taken by a large number of students to complete a test has probability density function h, as shown in the diagram.

[ 4 ]

(i)

Without calculation, use the diagram to explain how you can tell that the median time is less than 15 minutes.

It is now given that

h(x)= \begin{cases}\frac{40}{x^{2}}-\frac{1}{10} & 10 \leqslant x \leqslant 20 \\ 0 & \text { otherwise. }\end{cases}

[ 1 ]

(ii)

Find the mean time.

[ 3 ]

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question 2

Question (a)

Question (b)

Question (c)

Question (i)

Question (ii)