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