Chapter 5 - Joint Probability Distributions and Random Samples
SHORT ANSWER
1. Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 26 psi. Suppose the actual air
pressure in each tire is a random variable—X for the right tire and Y for the left tire, with joint pdf
a. What is the value of K?
b. What is the probability that both tires are underfilled?
c. What is the probability that the difference in air pressure between the two tires is at most 2 psi?
d. Determine the (marginal) distribution of air pressure in the right tire alone.
e. Are X and Y independent random variables?
ANS:
a.
, b.
c.
=
=
= .3593 (after much algebra)
d.
e. is obtained by substituting y for x in (d); clearly are
not independent.
PTS: 1
2. Let X denote the number of brand X VCRs sold during a particular week by a certain store. The pmf of X is
x 0 1 2 3 4
.1 .2 .3 .25 .15
Seventy percent of all customers who purchase brand X VCRs also buy an extended warranty. Let Y denote the
number of purchasers during this week who buy an extended warranty.
a. What is P(X = 4, Y = 2)? [Hint: This probability equals P(Y = 2/X = 4) P(X = 4); now think of the four
purchases as four trials of a binomial experiment, with success on a trial corresponding to buying an extended
warranty.]
b. Calculate P(X =Y).
c. Determine the joint pmf of X and Y and then the marginal pmf of Y.
ANS:
a.
b.
c. For any such pair,
, PTS: 1
3. Two components of a minicomputer have the following joint pdf for their useful lifetimes X and Y:
a. What is the probability that the lifetime X of the first component exceeds 3?
b. What are the marginal pdf”s of X and Y? Are the two lifetimes independent? Explain.
c. What is the probability that the lifetime of at least one component exceeds 3?
ANS:
a.
b. The marginal pdf of X is
It is now clear that f(x,y) is not the product of the marginal pdf”s, so the two
random variables are not independent.
c.
PTS: 1
4. The joint pdf of pressures for right (X) and left (Y) front tires is given by
.
a. Determine the conditional pdf of Y given that X = x and the conditional pdf of X given that Y = y if you are
given
b. If the pressure in the right tire is found to be 22 psi, what is the probability that the left tire has a pressure of at
least 25 psi? Compare this to
SHORT ANSWER
1. Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 26 psi. Suppose the actual air
pressure in each tire is a random variable—X for the right tire and Y for the left tire, with joint pdf
a. What is the value of K?
b. What is the probability that both tires are underfilled?
c. What is the probability that the difference in air pressure between the two tires is at most 2 psi?
d. Determine the (marginal) distribution of air pressure in the right tire alone.
e. Are X and Y independent random variables?
ANS:
a.
, b.
c.
=
=
= .3593 (after much algebra)
d.
e. is obtained by substituting y for x in (d); clearly are
not independent.
PTS: 1
2. Let X denote the number of brand X VCRs sold during a particular week by a certain store. The pmf of X is
x 0 1 2 3 4
.1 .2 .3 .25 .15
Seventy percent of all customers who purchase brand X VCRs also buy an extended warranty. Let Y denote the
number of purchasers during this week who buy an extended warranty.
a. What is P(X = 4, Y = 2)? [Hint: This probability equals P(Y = 2/X = 4) P(X = 4); now think of the four
purchases as four trials of a binomial experiment, with success on a trial corresponding to buying an extended
warranty.]
b. Calculate P(X =Y).
c. Determine the joint pmf of X and Y and then the marginal pmf of Y.
ANS:
a.
b.
c. For any such pair,
, PTS: 1
3. Two components of a minicomputer have the following joint pdf for their useful lifetimes X and Y:
a. What is the probability that the lifetime X of the first component exceeds 3?
b. What are the marginal pdf”s of X and Y? Are the two lifetimes independent? Explain.
c. What is the probability that the lifetime of at least one component exceeds 3?
ANS:
a.
b. The marginal pdf of X is
It is now clear that f(x,y) is not the product of the marginal pdf”s, so the two
random variables are not independent.
c.
PTS: 1
4. The joint pdf of pressures for right (X) and left (Y) front tires is given by
.
a. Determine the conditional pdf of Y given that X = x and the conditional pdf of X given that Y = y if you are
given
b. If the pressure in the right tire is found to be 22 psi, what is the probability that the left tire has a pressure of at
least 25 psi? Compare this to
