4.139 The moment-generating function of a normally distributed random variable, μ and variance o² was shown in Exercise 4.138 to be m(t) et+ (1/2)²2. in Exercise 4.137 to derive the moment-generating distribution of X? Why? Y, with mean Use the result = function of X = -3Y + 4. What is the
4.139 The moment-generating function of a normally distributed random variable, μ and variance o² was shown in Exercise 4.138 to be m(t) et+ (1/2)²2. in Exercise 4.137 to derive the moment-generating distribution of X? Why? Y, with mean Use the result = function of X = -3Y + 4. What is the
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Pls how do i solve 4.139?
![4.136
4.139
4.140
I can be used as numerical descriptive measures to describe the data that we obtain in
The purpose of the preceding discussion of moments is twofold. First, moments
variable possesses a particular probability distribution. It can be shown that if two
an experiment. Second, they can be used in a theoretical sense to prove that a random
random variables Y and Z possess identical moment-generating functions, then Y
and Z possess identical probability distributions. This latter application of moments
was mentioned in the discussion of moment-generating functions for discrete random
variables in Section 3.9; it applies to continuous random variables as well.
For your convenience, the probability and density functions, means, variances,
and moment-generating functions for some common random variables are given in
Appendix 2 and inside the back cover of this text.
Exercises
Suppose that the waiting time for the first customer to enter a retail shop after 9:00 A.M. is a
random variable Y with an exponential density function given by
y > 0,
a
{6).
f(y) =
e-x/e
4.138 Example 4.16 derives the moment-generating function for Yu, where Y is normally dis-
tributed with mean μ and variance o².
elsewhere.
a Find the moment-generating function for Y.
b Use the answer from part (a) to find E(Y) and V (Y).
Show that the result given in Exercise 3.158 also holds for continuous random variables. That
is, show that, if Y is a random variable with moment-generating function m(t) and U is given
by U = aY + b, the moment-generating function of U is ebm (at). If Y has mean and
variance o², use the moment-generating function of U to derive the mean and variance of U.
Use the results in Example 4.16 and Exercise 4.137 to find the moment-generating function
for Y.
a m(t) = (1-4t)-².
b m(t)=1/(1-3.2t).
c m(t) = e-51+61²
b Differentiate the moment-generating function found in part (a) to show that E(Y) = μ and
V(Y) = 0².
The moment-generating function of a normally distributed random variable, Y, with mean
μ and variance o2 was shown in Exercise 4.138 to be m(t) = et+ (1/2)²o². Use the result
in Exercise 4.137 to derive the moment-generating function of X = -3Y + 4. What is the
distribution of X? Why?
Identify the distributions of the random variables with the following moment-generating
functions:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F22752cce-aae9-4c92-9942-5544ae988fbe%2F5e2639c9-009a-4ae5-9362-c5950ae9a669%2Fm453rwj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4.136
4.139
4.140
I can be used as numerical descriptive measures to describe the data that we obtain in
The purpose of the preceding discussion of moments is twofold. First, moments
variable possesses a particular probability distribution. It can be shown that if two
an experiment. Second, they can be used in a theoretical sense to prove that a random
random variables Y and Z possess identical moment-generating functions, then Y
and Z possess identical probability distributions. This latter application of moments
was mentioned in the discussion of moment-generating functions for discrete random
variables in Section 3.9; it applies to continuous random variables as well.
For your convenience, the probability and density functions, means, variances,
and moment-generating functions for some common random variables are given in
Appendix 2 and inside the back cover of this text.
Exercises
Suppose that the waiting time for the first customer to enter a retail shop after 9:00 A.M. is a
random variable Y with an exponential density function given by
y > 0,
a
{6).
f(y) =
e-x/e
4.138 Example 4.16 derives the moment-generating function for Yu, where Y is normally dis-
tributed with mean μ and variance o².
elsewhere.
a Find the moment-generating function for Y.
b Use the answer from part (a) to find E(Y) and V (Y).
Show that the result given in Exercise 3.158 also holds for continuous random variables. That
is, show that, if Y is a random variable with moment-generating function m(t) and U is given
by U = aY + b, the moment-generating function of U is ebm (at). If Y has mean and
variance o², use the moment-generating function of U to derive the mean and variance of U.
Use the results in Example 4.16 and Exercise 4.137 to find the moment-generating function
for Y.
a m(t) = (1-4t)-².
b m(t)=1/(1-3.2t).
c m(t) = e-51+61²
b Differentiate the moment-generating function found in part (a) to show that E(Y) = μ and
V(Y) = 0².
The moment-generating function of a normally distributed random variable, Y, with mean
μ and variance o2 was shown in Exercise 4.138 to be m(t) = et+ (1/2)²o². Use the result
in Exercise 4.137 to derive the moment-generating function of X = -3Y + 4. What is the
distribution of X? Why?
Identify the distributions of the random variables with the following moment-generating
functions:
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