26(a) Suppose X~ x² (m), S = X+Y~x² (m+n), and X and Y are independent. Use moment-generating functions to show that 5- X ~ x² (n).
26(a) Suppose X~ x² (m), S = X+Y~x² (m+n), and X and Y are independent. Use moment-generating functions to show that 5- X ~ x² (n).
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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Transcribed Image Text:Q6(a) Suppose X ~ x²(m), S = X+Y~x² (m+n), and X and Y are independent. Use moment-generating functions to show that
S-X~x² (n).
Q6(b) Let Y₁ ~ x² (n). The goal is to find the limiting distribution of
Zn =
(Yn-n)
√2n
as n → ∞. First, express the moment-generating function Mz, (t) of Zn in terms of the moment-generating function My, (t) of Yn. Then
find ln(Mz, (t). Use the fact In(1-x) = -x-x² /2-x³/3 - ²/4 to show that as n →∞, Mz, (t) converges to et²/2 and hence Zn
converges to N(0, 1) as n → ∞.
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