Problem 3. An interesting property of Student's & distribution is that the moment generating function does not exist since only finitely many moments exist (for the moment generating function to exist we would need infinitely many moments to exist). a) Once again, the density of the Student's & distribution with d degrees of freedom is given by -(d+1)/2 fr(t) = ² (₁+²) LER. r((d+1)/2) T(d/2)√dn Compute the expectation of T. Be careful to state for what restriction on powers (which will depend on the parameter d) this integral converges. b) Verify that your restriction from part a) implies that the expectation only exists for d > 1. e) Consider the nth moment of T for n > 0. Do not calculate this moment directly (which can be done, but is tedious!), but show for what powers the integral will converge. Note that only the large-t behavior will affect the convergence, so this can be done immediately! By part c), we can conclude that only the first finitely many moments exist, therefore the moment generating function does not exist for the distribution!
Problem 3. An interesting property of Student's & distribution is that the moment generating function does not exist since only finitely many moments exist (for the moment generating function to exist we would need infinitely many moments to exist). a) Once again, the density of the Student's & distribution with d degrees of freedom is given by -(d+1)/2 fr(t) = ² (₁+²) LER. r((d+1)/2) T(d/2)√dn Compute the expectation of T. Be careful to state for what restriction on powers (which will depend on the parameter d) this integral converges. b) Verify that your restriction from part a) implies that the expectation only exists for d > 1. e) Consider the nth moment of T for n > 0. Do not calculate this moment directly (which can be done, but is tedious!), but show for what powers the integral will converge. Note that only the large-t behavior will affect the convergence, so this can be done immediately! By part c), we can conclude that only the first finitely many moments exist, therefore the moment generating function does not exist for the distribution!
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
Related questions
Question
Thanks
Transcribed Image Text:Problem 3. An interesting property of Student's & distribution is that the moment generating
function does not exist since only finitely many moments exist (for the moment generating function
to exist we would need infinitely many moments to exist).
a) Once again, the density of the Student's & distribution with d degrees of freedom is given by
-(d+1)/2
fr(t) = ² (₁+²) LER.
r((d+1)/2)
T(d/2)√dn
Compute the expectation of T. Be careful to state for what restriction on powers (which will
depend on the parameter d) this integral converges.
b) Verify that your restriction from part a) implies that the expectation only exists for d > 1.
e) Consider the nth moment of T for n > 0. Do not calculate this moment directly (which can
be done, but is tedious!), but show for what powers the integral will converge. Note that
only the large-t behavior will affect the convergence, so this can be done immediately!
By part c), we can conclude that only the first finitely many moments exist, therefore the moment
generating function does not exist for the distribution!
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps
Recommended textbooks for you
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON