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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 1 fv(u) = /u³-¹e-³, u > 0

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 1 fv(u) = /u³-¹e-³, u > 0

A First Course in Probability (10th Edition)
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...
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Please show me the question of I and II, thank you so much 

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 k fv(u) = 1/2 u 2 e-²₂ u > 0 where 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 FT be the CDF of the random variable T. Verify that = 12 Z √U/k {1+P + P(Z² ≤ U) ¹P (Z² ≤ U) ift > 0, ift < 0. (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 ( 2² ≤ ²U) = [° F₂² (u) fv(u)du, < k where Fz2 is the CDF of Z2 and fu is the PDF of U.
Transcribed Image Text: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 k fv(u) = 1/2 u 2 e-²₂ u > 0 where 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 FT be the CDF of the random variable T. Verify that = 12 Z √U/k {1+P + P(Z² ≤ U) ¹P (Z² ≤ U) ift > 0, ift < 0. (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 ( 2² ≤ ²U) = [° F₂² (u) fv(u)du, < k where Fz2 is the CDF of Z2 and fu is the PDF of U.
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