STAT/MATH 511, Summer I, 2008 Exam I
This exam contains 5 questions; each question is worth 10 points. Print your name at
the top of this page in the upper right hand corner. You have 2 hours to complete this
exam. GOOD LUCK!!
1. Five identical bowls are labeled 1, 2, 3, 4, and 5. Bow icontains iwhite and 5 −i
black balls, with i= 1,2,3,4,5. A bowl is randomly selected and two balls are randomly
selected (without replacement) from the contents of the bowl.
(a) What is the probability that both balls selected are white?
(b) Given that both balls selected are white, what is the probability that bowl 3 was
selected?
2. Suppose that Yhas a poisson distribution with parameter λso that the pmf of Yis
P(Y=y) = (λye−λ
y!, y = 0,1,2, ...
0,otherwise.
(a)Prove that P∞
y=0 P(Y=y) = 1
(b)Find the moment-generating function for Y.
(c)Prove that the mean of Yis λ.
(d)Prove that the variance of Yis λ.
3. If Yhas a geometric distribution with success probability p,
(a) Find P(Y= an even integer)
(b) Define X=Y−1. If Yis interpreted as the number of the trial on which the
first success occurs, then Xcan be interpreted as the number of failures before the first
success. Find the pmf of X, the mean of X, and the variance of X.
4. A standard deck of cards contains 52 cards in four suits (clubs, diamonds, hearts, and
spades) and thirteen ranks running from two to ten, jack, queen, king, and ace. Five
cards are dealt at random without replacement from a standard deck of 52 cards. What
is the probability that
(a) the hand contains all spades?
(b) two cards are kings, two cards are queens and one card is jack.
(c) the hand contains all four aces if it is know that it contains at least three aces.
5. A random variable Yhas a logarithmic series distribution with parameter p if
P(Y=y) = (−(1−p)y
y·ln(p), y = 1,2, ..., 0< p < 1
0,otherwise.
(a)Prove that P∞
y=1 P(Y=y) = 1
(b)Find that the mean of Y,the variance of Y.
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