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Date May 2021 Marks available 2 Reference code 21M.2.AHL.TZ2.6
Level Additional Higher Level Paper Paper 2 Time zone Time zone 2
Command term Show that Question number 6 Adapted from N/A

Question

A continuous random variable X has the probability density function fn given by

fn(x)={(n+1)xn,     0,  0x1otherwise

where n, n0.

Show that E(X)=n+1n+2.

[2]
a.

Show that Var(X)=n+1(n+2)2(n+3).

[4]
b.

Markscheme

E(X)=(n+1)10xn+1dx                       M1

=(n+1)[xn+2n+2]10                         A1

leading to E(X)=n+1n+2                   AG

 

[2 marks]

a.

METHOD 1

use of Var(X)=E(X2)-[E(X)]2                  M1

Var(X)=(n+1)10xn+2dx-(n+1n+2)2

=(n+1)[1n+3xn+3]10-(n+1n+2)2

=n+1n+3-(n+1n+2)2                       A1

=(n+1)(n+2)2-(n+1)2(n+3)(n+2)2(n+3)                  M1


EITHER

=(n+1)(n2+4n+4-(n2+4n+3))(n+2)2(n+3)                       A1


OR

=(n3+5n2+8n+4)-(n3+5n2+7n+3)(n+2)2(n+3)                       A1


THEN

so Var(X)=n+1(n+2)2(n+3)                       AG

 

METHOD 2

use of Var(X)=E(X-E(X))2                  M1

Var(X)=(n+1)10(x-n+1n+2)2xndx

=(n+1)[1n+3xn+3-2(n+1)(n+2)2xn+2+n+1(n+2)2xn+1]10

=n+1n+3-(n+1n+2)2                       A1

=(n+1)((n+2)2-(n+1)(n+3))(n+2)2(n+3)                  M1


EITHER

=(n+1)(n2+4n+4-(n2+4n+3))(n+2)2(n+3)                       A1


OR

=(n3+5n2+8n+4)-(n3+5n2+7n+3)(n+2)2(n+3)                       A1


THEN

so Var(X)=n+1(n+2)2(n+3)                       AG

 

[6 marks]

b.

Examiners report

[N/A]
a.
[N/A]
b.

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

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