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Unformatted text preview: 1. (20 points) A coin is ﬂipped until there are two heads in a row, or three tails. Find the
probability that the three tails comes ﬁrst. "fﬁlﬂffmﬁnj 7L0, Fume/v» 0L5 VJYtHQh
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I; 1 06“ RN Z Acacb In Q row 0/4 gzéz 1A0 row H: 6%wa 11"“?ch in 0‘ r0L3 T: eA/Qn" 33(0)“ (ANVDLJ 2. (20 points) Three integers are chosen with replacement from the ﬁrst twenty integers. Find
the probability that a. (10 pts) Their sum is even SUM {I €U€n 1‘16 I] A” OWK Cl/‘Cn b. (10 pts) Their product is even. Paschal {s :2.th “ab? a“ 00(0) 3. (20 points) X is a continuous random variable With cdf FX(m) = 1 — (1 — :c)2 for 0 S x S l,
with FX(ac) = 0 for :3 g 0, and FX(a:) = 1 for m 2 1. a. (10 pts) Find the pdf of X. :CM‘ 5% FA] if? [*U—XJQ : 2m} f0, 0&4
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O 160/ ><<O\ b. (10 pts) Let Y = 2(X + 1)? Find the cdf of Y. 4. (20 points) A random integer N from 1 to 10 is Chosen uniformly at random. 21. (10 pts) The random variable X denotes the number of distinct prime factors of N.
(So if N— 8: 23X = 1). Find the pmfof X >5 P”) [6V0 {”wa QGLMS 7A0 lwqgqu? Vlyf'wkcgohr
Qitg a“ 2 [Prime Qintwg b. (10 pts) Another random integer N ’ from 1 to 10 is also chosen uniformly at random.
Find the probability that N > 2N ’ . @5le (NIN‘JT
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mum to s. 5. (20 points) IS it possible for two events A and B with 1P(A), ]P’(B) > 0 to be both mutually
exclusive and independent. Why or Why not? I? @ A K are, inerf CHJZQVJL “WW W WW8 (B)>
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 Fall '10
 Dr.PaulHorn

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