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Date May 2014 Marks available 16 Reference code 14M.3sp.hl.TZ0.4
Level HL only Paper Paper 3 Statistics and probability Time zone TZ0
Command term Find, Hence, Show that, and State Question number 4 Adapted from N/A

Question

Consider the random variable XGeo(p)XGeo(p).

(a)     State P(X<4)P(X<4).

(b)     Show that the probability generating function for X is given by GX(t)=pt1qtGX(t)=pt1qt, where q=1pq=1p.

Let the random variable Y=2XY=2X.

(c)     (i)     Show that the probability generating function for Y is given by GY(t)=GX(t2)GY(t)=GX(t2).

          (ii)     By considering GY(1), show that E(Y)=2E(X).

Let the random variable W=2X+1.

(d)     (i)     Find the probability generating function for W in terms of the probability generating function of Y.

          (ii)     Hence, show that E(W)=2E(X)+1.

Markscheme

(a)     use of P(X=n)=pqn1 (q=1p)     (M1)

P(X<4)=p+pq+pq2 (=1q3) (=1(1p)3) (=3p3p2+p3)     A1

[2 marks]

 

(b)   GX(t)=P(X=1)t+P(X=2)t2+     (M1)

=pt+pqt2+pq2t3+     A1

summing an infinite geometric series     M1

=pt1qt     AG

[3 marks]

 

(c)     (i)     EITHER

          GY(t)=P(Y=1)t+P(Y=2)t2+     A1

          =0×t+P(X=1)t2+0×t3+P(X=2)t4+     M1A1

          =GX(t2)     AG

          OR

          GY(t)=E(tY)=E(t2X)     M1A1

          =E((t2)X)     A1

          =GX(t2)     AG

          (ii)     E(Y)=GY(1)     A1

          EITHER

          =2tGX(t2) evaluated at t=1     M1A1

          =2E(X)     AG

          OR

          =ddx(pt2(1qt2))=2pt(1qt2)+2pqt3(1qt2)2 evaluated at t=1     A1

          =2×p(1qt)+pqt(1qt)2 evaluated at t=1 (or 2p)     A1

          =2E(X)     AG

[6 marks]

 

(d)     (i)     GW(t)=tGY(t) (or equivalent)     A2

          (ii)     attempt to evaluate GW(t)     M1

          EITHER

          obtain 1×GY(t)+t×GY(t)     A1

          substitute t=1 to obtain 1×1+1×GY(1)     A1

          OR

          =ddx(pt3(1qt2))=3pt2(1qt2)+2pqt4(1qt2)2     A1

          substitute t=1 to obtain 1+2p     A1

          =1+2E(X)     AG

[5 marks]

 

Total [16 marks]

Examiners report

[N/A]

Syllabus sections

Topic 7 - Option: Statistics and probability » 7.1 » Geometric distribution.

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