Question
1) A city commissioner claims that 80% of the people living in the city favour contracting out garbage collection to a private company. To test
1) A city commissioner claims that 80% of the people living in the city favour contracting
out garbage collection to a private company. To test the commissioners claim, 25
city residents are randomly selected, yielding 22 who prefer contracting to a private
company.
(a) If the commissioners claim is correct, what is the probability that the sample
would contain at least 22 who prefer contracting to a private company?
(b) If the commissioners claim is correct, what is the probability that exactly 22
would prefer contracting to a private company?
(c) Based on observing 22 in a sample of size 25 who prefer contracting to a private
company, what do you conclude about the commissioners claim that 80% of city
residents prefer privatization of garbage collection?
2) Of a population of consumers, 60% are reputed to prefer a particular brand, A, of toothpaste. If a group of randomly selected consumers is interviewed, what is the probability
that exactly ve people have to be interviewed to encounter the rst consumer who
prefers brand A?
3) Suppose that Y is a binomial random variable based on n trails with success probability
p and consider Z = n Y .
(a) Argue that for z = 0, 1, . . . , n
P (Z = z) = P (n Y = z) = P (Y = n z).
(b) Use the result from part (a) to show that
P (Z = z) =
n
pnz q z =
nz
n z nz
q p .
z
(c) The result in part (b) implies that Z is a binomial r.v. based on n trails and
success probability p = q = 1 p. Explain why this result is obvious.
4) If Y is a geometric r.v. as dened in lecture, let Z = Y 1. If Y is interpreted as
the trail on which the rst success occurs, then Z can be interpreted as the number
of failures before the rst success. If Z = Y 1, P (Z = y) = P (Z = y + 1) for
y = 0, 1, 2, . . . . Show that
P (Z = y) = q y p, y = 0, 1, 2, . . .
and derive the mean and variance of Z.
5) Consider the negative binomial distribution as dened in lecture.
(a) Show that if y r + 1,
P (Y =y)
P (Y =y1)
=
y1
yr
q. This establishes a recursive relation-
ship between successive probabilities since P (Y = y) = P (Y = y 1)
y1
yr
q.
STAT 2655
(b) Show that
if y >
P (Y =y)
P (Y =y1)
=
y1
yr
q > 1 if y <
r1
.
1q
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