Unformatted text preview: Stat 116 Final Examination
Show all work ! Justify your assertions ! 1. (5 p0ints+5+5+5) For random variables X and Y with joint density function and f (:13, y) = 0 otherwise, ﬁnd
(a) P(X g 2,Y g 3).
(b) fx($) (C) fy(y)
(d) Are X and Y independent? Explain. . (15) A hat contains n coins, f of which are fair, and b of which are biased to land with heads
with probability g, with f + b = n. A coin is drawn at random from the hat and tossed
once. It lands heads. What is the probability that it is a biased coin?. . (10+10) Particles arrive at a Geiger counter according to a Poisson process with rate 3 per minute.
(a) Find the chance that less than 4 particles arrive in the time interval 0 to 2 minutes.
(b) Let Tn minutes denote the arrival time of the nth particle. Find P(T1<1,T2—T1 <1,T3—T2<1) . (10+10) Let R and S be independently exponentially distributed random variables with parame—
ters /\ and [1 respectively. (a) Find the density function for M = min(R, S). (Hint: Find P(M 2 23)) (b) Calculate P(R 2 S). . . (15) Let X, Y be independent, standard normal random variables. Let A = {(x, y) E R2 : 9 <
2:2 + y2 < 16}. Find P((X, Y) E A). (A numerical answer is requested.) . (7+7+7+7) Cards from a well shufﬂed ordinary deck are turned face up one at a time. Let W1 =
number of cards before the ﬁrst ace, and let W5 = number of cards after the last (fourth)
ace. For i = 2, 3, 4, let IV, = number of cards between the (2'  1)th and the ith ace. (Of
course, any W; may possibly be 0).
(a) Let A = {($1,x2,x3,x4,$5)l each 1:, is an integer Z 0 and 2,1135, = 48}. Find the
number of elements in the set A.
(b) Find P(Wi = 101, W2 = $2, W3 = 133, W4 = 154, W5 _= $5) 1 ...
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 Summer '07
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 Probability

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