Homework Help is Here – Start Your Trial Now!
Math
Statistics
Let U~ x²(k), a chi-squared distribution with k degrees of freedom. In lecture, we derived the form of the PDF of U to be fv(u) = u-¹e-‡, u>0 where Z is the normalization constant making this function a PDF. Now, let Z ~ N(0,1) be independent from U, and define Z √U/k Then Tt(k), a t distribution with k degrees of freedom. In this exercise, you will derive the form of the PDF of T following the same type of calculations in lecture. (i) Let Fr be the CDF of the random variable T. Verify that Fr(t) = T = + PZ² 0, ift < 0. fr(t) = √fz² (zu) · 21 u · ƒv(u)du, where fr is the PDF of T and fz2 is the PDF of Z². (iv) Plug in the PDFs of Z² and U to get that fou Jo • (-u) fv (u)du, fr(t) = constant x fr(t) = constant x u-¹e-1 (¹+²) = du. (Hint: Z² ~ x²(1).) (v) Apply the change of variable v = = (1+²) u to get that (1 + ²/²)

Let U~ x²(k), a chi-squared distribution with k degrees of freedom. In lecture, we derived the form of the PDF of U to be fv(u) = u-¹e-‡, u>0 where Z is the normalization constant making this function a PDF. Now, let Z ~ N(0,1) be independent from U, and define Z √U/k Then Tt(k), a t distribution with k degrees of freedom. In this exercise, you will derive the form of the PDF of T following the same type of calculations in lecture. (i) Let Fr be the CDF of the random variable T. Verify that Fr(t) = T = + PZ² 0, ift < 0. fr(t) = √fz² (zu) · 21 u · ƒv(u)du, where fr is the PDF of T and fz2 is the PDF of Z². (iv) Plug in the PDFs of Z² and U to get that fou Jo • (-u) fv (u)du, fr(t) = constant x fr(t) = constant x u-¹e-1 (¹+²) = du. (Hint: Z² ~ x²(1).) (v) Apply the change of variable v = = (1+²) u to get that (1 + ²/²)

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
See similar textbooks
icon
Related questions
Question

part iii and iv
got parts i and ii answered previously

Exercise 6 Let U x²(k), a chi-squared distribution with k degrees of freedom. In lecture, we derived the form of the PDF of U to be fv(u) = /u-¹e, u>0 where Z is the normalization constant making this function a PDF. Now, let Z ~ N(0,1) be independent from U, and define Fr(t) = T = Then T t(k), a t distribution with k degrees of freedom. In this exercise, you will derive the form of the PDF of T following the same type of calculations in lecture. (i) Let Fr be the CDF of the random variable T. Verify that Z √U/k < [ + P(Z² < £U) - P(Z² ≤ U) (Hint: consider the event |Z| ≤ t√U/k and draw a picture of the PDF of Z.) (ii) Use the Law of Total Probability to show that P(2²sU)-F₂(x) fv(n)du, = where Fz² is the CDF of Z2 and fu is the PDF of U. (iii) Differentiate the integral to get that fr(t) = 1/2 √² fz² • (1/₁). fr(t) = constant x 2|t| k where fr is the PDF of T and fz2 is the PDF of Z². (iv) Plug in the PDFs of Z² and U to get that ift > 0, ift < 0. -U. (Hint: Z²x²(1).) (v) Apply the change of variable v = · ( ¹ + ² ) u fu(u)du, √ ² u²+ - ¹ 6 - ² (1+² ) " du. U Jo u to get that fr(t) = constant x + (Hint: everything that doesn't involve t can be dumped into the constant factor.)
Transcribed Image Text:Exercise 6 Let U x²(k), a chi-squared distribution with k degrees of freedom. In lecture, we derived the form of the PDF of U to be fv(u) = /u-¹e, u>0 where Z is the normalization constant making this function a PDF. Now, let Z ~ N(0,1) be independent from U, and define Fr(t) = T = Then T t(k), a t distribution with k degrees of freedom. In this exercise, you will derive the form of the PDF of T following the same type of calculations in lecture. (i) Let Fr be the CDF of the random variable T. Verify that Z √U/k < [ + P(Z² < £U) - P(Z² ≤ U) (Hint: consider the event |Z| ≤ t√U/k and draw a picture of the PDF of Z.) (ii) Use the Law of Total Probability to show that P(2²sU)-F₂(x) fv(n)du, = where Fz² is the CDF of Z2 and fu is the PDF of U. (iii) Differentiate the integral to get that fr(t) = 1/2 √² fz² • (1/₁). fr(t) = constant x 2|t| k where fr is the PDF of T and fz2 is the PDF of Z². (iv) Plug in the PDFs of Z² and U to get that ift > 0, ift < 0. -U. (Hint: Z²x²(1).) (v) Apply the change of variable v = · ( ¹ + ² ) u fu(u)du, √ ² u²+ - ¹ 6 - ² (1+² ) " du. U Jo u to get that fr(t) = constant x + (Hint: everything that doesn't involve t can be dumped into the constant factor.)
Expert Solution
steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman
SEE MORE TEXTBOOKS