Krzys’ Ostaszewski: http://www.krzysio.net
Author of the BTDT Manual (the “Been There Done That!” manual) for Course P/1
http://smartURL.it/krzysioP (paper) or http://smartURL.it/krzysioPe (electronic)
Instructor of online P/1 seminar: http://smartURL.it/onlineactuary
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Dr. Ostaszewski’s online exercise posted October 16, 2010
1
. Let  x  be the
λ
greatest integer function, denoting the greatest integer among those not exceeding x.
Which of the following is the correct expression for the expected value of N =  X ?
A random variable X has the exponential distribution with mean
1
A.  
λ 
1 1
B.   −
λ  2
C.
1
1
−
λ  2
D.
eλ
eλ − 1
E.
1
e −1
λ
Solution.
Note that N is a discrete non-negative random variable, so that its expected value can be
calculated as:
+∞
+∞
+∞
+∞
n =1
n =1
n =1
n =1
E ( N ) = ∑ Pr ( N ≥ n ) = ∑ Pr ( X  ≥ n ) = ∑ Pr ( X ≥ n ) = ∑ e− λ n =
e− λ
1
= λ
.
−λ
1− e
e −1
Answer E.
© Copyright 2004-2010 by Krzysztof Ostaszewski.
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Exercises from the past actuarial examinations are copyrighted by the Society of
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