Date | May 2017 | Marks available | 4 | Reference code | 17M.3sp.hl.TZ0.3 |
Level | HL only | Paper | Paper 3 Statistics and probability | Time zone | TZ0 |
Command term | Hence and Determine | Question number | 3 | Adapted from | N/A |
Question
The discrete random variable X has the following probability distribution.
P(X=x)={pqx2for x=0, 2, 4, 6… where p+q=1, 0<p<1.0otherwise
Show that the probability generating function for X is given by G(t)=P1−qt2.
Hence determine E(X) in terms of p and q.
The random variable Y is given by Y=2X+1. Find the probability generating function for Y.
Markscheme
G(t)=∑P(X=x)tx (M1)
=p+pqt2+pq2t4+…
(summing GP) u1=p, r=qt2 A1
=p1−qt2 AG
[2 marks]
G′(t)=−p(1−qt2)2×−2qt M1A1
E(X)=G′(1) (M1)
=2pq(1−q)2(=2qp) A1
[4 marks]
METHOD 1
PGF of Y=∑P(Y=y)ty (M1)
=pt+pqt5+pq2t9+… A1
=pt1−qt4 A1
METHOD 2
PGF of Y=E(tY) (M1)
=E(t2X+1)
=E((t2)X)×E(t) A1
=pt1−qt4 A1
[3 marks]