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5.5-15. Let the distribution of T be t(17). Find (a) 10.01 (17). (b) t9,95(17). (c) P(-1.740

5.5-15. Let the distribution of T be t(17). Find (a) 10.01 (17). (b) t9,95(17). (c) P(-1.740

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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5.5-15

**Educational Transcription:** **Section 5.5.14** - Let \( T \) have a distribution with \( r \) degrees of freedom. Show that \( E(T) = 0 \) provided that \( r = 2 \). - (b) Show that \( \text{Var}(T) = r/(r - 2) \) provided that \( r > 2 \). - (c) Let the distribution of \( T \) be \( t(17) \). Find \( P(-1.740 \leq T \leq 1.740) \). **Details on \( V \) and \( W \):** - \( W = \frac{Z_1}{\sqrt{(Z_2 + Z_3)/2}} \) - \( V = \frac{Z_1}{\sqrt{(Z_2 + Z_3)/2}} \) **Properties of \( V \):** - Has pdf \( f(v) = 1/(\sqrt{\pi}(2 - v^2)) \), \( -\sqrt{2} < v < \sqrt{2} \). - (c) Find the mean of \( V \). - (d) Find the standard deviation of \( V \). **Questions:** - (e) Why are the distributions of \( W \) and \( V \) so different? **Section 5.5.15** - (a) Find \( t_{0.05}(17) \). - (b) Solve the inequality \( -t_{0.025} \leq T \leq t_{0.025} = 0.95 \). - (c) Find \( t_{0.025} \) so that \( P(-t_{0.025} \leq T \leq t_{0.025}) = 0.85 \). **Section 5.5.16** - Let \( n = 9 \) in \( T \) as defined in Equation 5.5.2. - (a) Find \( t_{0.025} \) so that \( P(-t_{0.025} \leq T \leq t_{0.025}) = 0.95 \).
Transcribed Image Text:**Educational Transcription:** **Section 5.5.14** - Let \( T \) have a distribution with \( r \) degrees of freedom. Show that \( E(T) = 0 \) provided that \( r = 2 \). - (b) Show that \( \text{Var}(T) = r/(r - 2) \) provided that \( r > 2 \). - (c) Let the distribution of \( T \) be \( t(17) \). Find \( P(-1.740 \leq T \leq 1.740) \). **Details on \( V \) and \( W \):** - \( W = \frac{Z_1}{\sqrt{(Z_2 + Z_3)/2}} \) - \( V = \frac{Z_1}{\sqrt{(Z_2 + Z_3)/2}} \) **Properties of \( V \):** - Has pdf \( f(v) = 1/(\sqrt{\pi}(2 - v^2)) \), \( -\sqrt{2} < v < \sqrt{2} \). - (c) Find the mean of \( V \). - (d) Find the standard deviation of \( V \). **Questions:** - (e) Why are the distributions of \( W \) and \( V \) so different? **Section 5.5.15** - (a) Find \( t_{0.05}(17) \). - (b) Solve the inequality \( -t_{0.025} \leq T \leq t_{0.025} = 0.95 \). - (c) Find \( t_{0.025} \) so that \( P(-t_{0.025} \leq T \leq t_{0.025}) = 0.85 \). **Section 5.5.16** - Let \( n = 9 \) in \( T \) as defined in Equation 5.5.2. - (a) Find \( t_{0.025} \) so that \( P(-t_{0.025} \leq T \leq t_{0.025}) = 0.95 \).
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