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Date November 2015 Marks available 2 Reference code 15N.3sp.hl.TZ0.4
Level HL only Paper Paper 3 Statistics and probability Time zone TZ0
Command term Find and Hence or otherwise Question number 4 Adapted from N/A

Question

A discrete random variable U follows a geometric distribution with p=14.

Find F(u), the cumulative distribution function of U, for u=1, 2, 3

[3]
a.

Hence, or otherwise, find the value of P(U>20).

[2]
b.

Prove that the probability generating function of U is given by Gu(t)=t43t.

[4]
c.

Given that UiGeo(14), i=1, 2, 3, and that V=U1+U2+U3, find

(i)     E(V);

(ii)     Var(V);

(iii)     Gv(t), the probability generating function of V.

[6]
d.

A third random variable W, has probability generating function Gw(t)=1(43t)3.

By differentiating Gw(t), find E(W).

 

[4]
e.

A third random variable W, has probability generating function Gw(t)=1(43t)3.

Prove that V=W+3.

[3]
f.

Markscheme

METHOD 1

P(U=u)=14(34)u1     (M1)

F(u)=P(Uu)=ur=114(34)r1(or equivalent)

=14(1(34)u)134     (M1)

=1(34)u     A1

METHOD 2

P(Uu)=1P(U>u)     (M1)

P(U>u)= probability of u consecutive failures     (M1)

P(Uu)=1(34)u     A1

[3 marks]

a.

P(U>20)=1P(U20)     (M1)

=(34)20(=0.00317)     A1

[2 marks]

b.

GU(t)=r=114(34)r1tr(or equivalent)     M1A1

=r=113(34t)r     (M1)

=13(34t)134t(=14t134t)     A1

=t43t     AG

[4 marks]

c.

(i)     E(U)=114=4     (A1)

E(U1+U2+U3)=4+4+4=12     A1

(ii)     Var(U)=34(14)2=12     A1

Var(U1+U2+U3)=12+12+12=36     A1

(iii)     Gv(t)=(GU(t))3     (M1)

=(t43t)3     A1

[6 marks]

d.

GW(t)=3(43t)4(3)(=9(43t)4)     (M1)(A1)

E(W)=GW(1)=9     (M1)A1

 

Note:     Allow the use of the calculator to perform the differentiation.

[4 marks]

e.

EITHER

probability generating function of the constant 3 is t3     A1

OR

GW3(t)=E(tW+3)=E(tW)E(t3)     A1

THEN

W+3 has generating function GW+3=1(43t)3×t3=GV(t)     M1

as the generating functions are the same V=W+3     R1AG

[3 marks]

Total [22 marks]

f.

Examiners report

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Syllabus sections

Topic 7 - Option: Statistics and probability » 7.1 » Cumulative distribution functions for both discrete and continuous distributions.
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