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Unformatted text preview: 1. (20 points) a. (10 pts) 10 people are in a room. Assuming their birthdays are
independent and uniformly distributed, What is the probability that at least two of them share
a birthday. b. (10 pts) How many people must be in a room so that the probability that at least two
of them share a birthday is 0.5? 2. (20 points) Recall: The mode of a random variable X is the value of m Where the pmf 10(30)
or pdf fX(a:) is maximized (depending on Whether the random variable is continuous.) Find the
mode of the following random variables. a. (8 pts) X is discrete with pmf 13(17) 2 6—101??? for a: = O, 1, 2, 3, . . .. 6°20” to pad may WM
“Hy. A" 1:; ch
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Pﬂxl ‘ b. (7pts) X is continuous With pdf fx 2 in :0), 0 g m g 7r fX(m) = 0 otherwise. oLPOM
if (A): zero oi X117?
dx K 2’ c. (71%) X is continuous with pdf fX(as) = e—i—le‘” for 0 3 ac _<_ 1, fX(a:) = 0 otherwise. .1. x x  a
Q“! C m mcmﬂﬁ. Mocﬂé l‘V‘ 19/) 3. (20 points) a. (10 pts) Construct an example showing that 3 events can be pairwise
independent, but not mutually independent. é FroM (1063':
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l / ' if b. (10 pts) Suppose A, B, and C are mutually independent with MA) = 1/2, MB) 2 1/4, Me) = 3/8. Find
P((A u BC) m CC)
Wanna/)6): 1W Ave“) W C
. : {mummy}WWW) W l I i '5
c(—:+%~%><%\z%q 4. (20 points) A random word is chosen unformly from the sentence ”HOW much wood could a~woodchuck chuck if a woodchuck could chuck wood.”
l l ‘1 )
Let X denote the length of the word.
a. (10 pts) Find the pmf of X V x £60 b. (10 pts) Suppose two words are chosen uniformly from the sentence, with replacement.
What is the probability that they have the same length? 'Pl‘matll" (ﬁll *féil+r%sll+(%1't+(%)’t W
J /l Note: Ulbl Mama“! {5 ‘
(are + (er) + W731 + (are)
7‘ fl i‘ fl
tow mm was ML ‘1 5. (20 points) a. (10 pts) Suppose X is discrete with pmf p(a:) = %  wig, p($) = 0 otherwise. Find the pmf on=X2. r I . I 1 «L ‘ 1
HM W'Lue ijJ; m gx I) ’3)“ b. (10 pts) Suppose X if continuous With pdf f 2 51; for m 2 1, f = 0 otherwise.
Find the pdf of Y 2 X2. SUM? (3%) Cf)”; )CYKj]: ’V 5:} 37‘ ...
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 Fall '10
 Dr.PaulHorn

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