http://www.math.ilstu.edu/krzysio/
Krzys’ Ostaszewski
Author of the BTDT Manual for Course P/1
available at http://smartURL.it/krzysioP or http://smartURL.it/krzysioPe
Instructor for online Course P/1 seminar: http://smartURL.it/onlineactuary
Exercise for July 23, 2005
Let X be a discrete random variable with moment generating function
1
1 +" t n
10t
M X (t ) = 1 + e + ! # .
4
2 n = 0 n!
for !" < t < +". Find Pr ( X ! 3) .
(
A.
1
8
B.
1
4
)
C.
1
2
D.
3
4
E. Cannot be determined
Solution.
When a random variable X is discrete and assumes values x1 , x2 , x3 ,… with probabilities
p1 , p2 , p3 ,… then its moment generating function is
M X ( t ) = p1 ! etx1 + p2 ! etx2 + p3 ! etx3 + ….
In this case, we have
1
1 +" t n 1
1
1
1
1
1
M X ( t ) = 1 + e10t + ! # = ! e0!t + ! e10!t + ! et = ! e0!t + ! e1!t + ! e10!t .
4
2 n = 0 n! 4
4
2
4
2
4
Therefore, the random variable whose MGF we are analyzing assumes values
and
x1 = 0,
x2 = 1,
x3 = 10,
with probabilities
1
1
1
p1 = ,
p2 = , and
p3 = ,
4
2
4
respectively, so that
1
Pr ( X ! 3) = Pr ( X = 10 ) = .
4
Answer B.
(
)
© Copyright 2005 by Krzysztof Ostaszewski.
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