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Date	May 2018	Marks available	7	Reference code	18M.2.AHL.TZ1.H_10
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Find	Question number	H_10	Adapted from	N/A

Question

The continuous random variable X has probability density function $f$ given by

$f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {3ax}&,&{0 \leqslant x < 0.5} \\ {a\left( {2 - x} \right)}&,&{0.5 \leqslant x < 2} \\ 0&,&{{\text{otherwise}}} \end{array}} \right.$

Show that $a = \frac{2}{3}$ .

[3]

a.

Find ${\text{P}}\left( {X < 1} \right)$ .

[3]

b.

Given that ${\text{P}}\left( {s < X < 0.8} \right) = 2 \times {\text{P}}\left( {2s < X < 0.8} \right)$ , and that 0.25 < s < 0.4 , find the value of s.

[7]

c.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$a\left[ {\int_0^{0.5} {3x\,{\text{d}}x} + \int_{0.5}^2 {\left( {2 - x} \right)} \,{\text{d}}x} \right] = 1$ M1

Note: Award the M1 for the total integral equalling 1, or equivalent.

$a\left( {\frac{3}{2}} \right) = 1$ (M1)A1

$a = \frac{2}{3}$ AG

[3 marks]

a.

EITHER

$\int_0^{0.5} {2x\,{\text{d}}x} + \frac{2}{3}\int_{0.5}^1 {\left( {2 - x} \right)} \,{\text{d}}x$ (M1)(A1)

$= \frac{2}{3}$ A1

OR

$\frac{2}{3}\int_1^2 {\left( {2 - x} \right)} \,{\text{d}}x = \frac{1}{3}$ (M1)

so ${\text{P}}\left( {X < 1} \right) = \frac{2}{3}$ (M1)A1

[3 marks]

b.

${\text{P}}\left( {s < X < 0.8} \right) = \int_s^{0.5} {2x\,{\text{d}}x} + \frac{2}{3}\int_{0.5}^{0.8} {\left( {2 - x} \right)} \,{\text{d}}x$ M1A1

$= \left[ {{x^2}} \right]_s^{0.5} + 0.27$

$0.25 - {s^2} + 0.27$ (A1)

${\text{P}}\left( {2s < X < 0.8} \right) = \frac{2}{3}\int_{2s}^{0.8} {\left( {2 - x} \right)} \,{\text{d}}x$ A1

$= \frac{2}{3}\left[ {2x - \frac{{{x^2}}}{2}} \right]_{2s}^{0.8}$

$\frac{2}{3}\left( {1.28 - \left( {4s - 2{s^2}} \right)} \right)$

equating

$0.25 - {s^2} + 0.27 = \frac{4}{3}\left( {1.28 - \left( {4s - 2{s^2}} \right)} \right)$ (A1)

attempt to solve for s (M1)

s = 0.274 A1

[7 marks]

c.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.

Syllabus sections

Topic 4—Statistics and probability » AHL 4.14—Properties of discrete and continuous random variables

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Topic 4—Statistics and probability