EC 233 - PS 3 – Chapter 3
(Probability Distribution of Discrete R.V’s; Mean and Variance; Binomial Distribution)
0.
a)
b)
c)
d)
Tell whether each of the following is a discrete or a continuous random variable.
The number of beers sold at a bar during a particular week.
The length of time it takes a person to drive 50 miles.
The interest rate on 3-month Treasury bills
The number of products returned to a store on a particular day.
1) (Textbook 3.10). A rental agency, which leases heavy equipment by the day, has found that
one expensive piece of equipment is leased, on the average, only one day in five. If rental on
one day is independent of rental on any other day, find the probability distribution of Y, the
number of days between a pair of rentals.
2) (Textbook 3.11) Persons entering a blood bank are such that 1 in 3 has type O+ blood and 1 in
15 has type O- blood. Consider three randomly selected donors for the blood bank. Let X
denote the number of donors with O+ blood and Y denote the number with O- blood. Find the
probability distributions for X and Y. Also find the probability distribution for X+Y, the
number of donors who have type O blood.
3) (Textbook 3.12). Let Y be a random variable with p(y) given below.
Find E(Y), E (1/Y) and E (Y2 -1)
Y
p(y)
1
0.4
2
0.3
3
0.2
4
0.1
4) (Textbook 3. 19). Who is the king of late night TV? An internet survey estimates that, when
given a choice between David Letterman and Jay Leno, 52% of the population prefers to
watch Jay Leno. Three late night TV watchers are randomly selected and asked which of the
two talk show hosts they prefer.
a) Find the probability distribution for Y, the number of viewers in the sample who prefer
Leno.
b) Construct a probability histogram for p(y)
c) What is the probability that exactly one of the three viewers prefers Leno?
d) What are the mean and standard deviation for Y?
e) What is the probability that the number of viewers favoring Leno falls within 3 standard
deviations of the mean?
5) (Textbook 3.23) In a gambling game a person draws a single card from an ordinary 52 playing
card deck. A person is paid $15 for drawing a jack (vale) or a queen (kız) and $5 for drawing a
king or an ace. A person who draws any other card pay $4. If a person plays this game, what
is the expected gain?
6) (Textbook 3.30) Suppose that Y is a discrete random variable with mean E(Y) = µ and variance
V(Y) = σ 2 and let X=Y+1
a) Do you expect the mean of X to be larger than, smaller than, or equal to µ = E(Y)? Why?
b) Express E (X) in terms of µ. Does this result agree with your answer to part a)
c) Recalling that the variance is a measure of spread, do you expect the variance of X to be
larger than, smaller than or equal to V(Y) = σ 2
d) Show that X = Y+1 and Y have equal variances.
7) (Textbook 3.33). Let Y be a discrete random variable with mean µ and variance σ 2 . If a and b
are constants prove that
E (aY +b) = a E(Y) + b
V(aY +b) = a2 V(Y) = a2 σ 2
8) (Textbook 3.40) The probability that a patient recovers from a stomach disease is 0.8.
Suppose 20 people are known to have contracted the disease. What is the probability that
a) Exactly 14 recover
b) At least 10 recover
c) At least 14 but not more than 18 recover
d) At most 16 recover
9) (Textbook 3.59)Ten motors are packaged for sale in a certain warehouse. The motors sell for
$100 each, but a “double your money back” guarantee is in effect (yani $200) for any
defectives the purchaser may receive. Find the expected net gain for the seller if the
probability of any one motor being defective is 0.08 (Assume that the quality of any one
motor is independent of that of the others).