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7-16. If P and -1 denote the distribution and quantile functions, respec- tively, for a standard normal random variable, what are the distribution and quantile functions for a N(u, o2) random variable? 7-17. Suppose U is a Unif(0, 1) random variable, and suppose F is a dis- tribution function and G the corresponding quantile function. Show that the random variable G(U) has F as its DF. Hint: for 0 < u <1 show that G(u) < x if and only if F(x) > u. This requires use of the fact that DF are right continuous.

7-16. If P and -1 denote the distribution and quantile functions, respec- tively, for a standard normal random variable, what are the distribution and quantile functions for a N(u, o2) random variable? 7-17. Suppose U is a Unif(0, 1) random variable, and suppose F is a dis- tribution function and G the corresponding quantile function. Show that the random variable G(U) has F as its DF. Hint: for 0 < u <1 show that G(u) < x if and only if F(x) > u. This requires use of the fact that DF are right continuous.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 32E
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7-16. If and -1 denote the distribution and quantile functions, respec- tively, for a standard normal random variable, what are the distribution and quantile functions for a N(u, o²) random variable? 7-17. Suppose U is a Unif(0, 1) random variable, and suppose F is a dis- tribution function and G the corresponding quantile function. Show that the random variable G(U) has F as its DF. Hint: for 0 < u < 1 show that G(u) < x if and only if F(x) > u. This requires use of the fact that DF are right continuous.
Transcribed Image Text:7-16. If and -1 denote the distribution and quantile functions, respec- tively, for a standard normal random variable, what are the distribution and quantile functions for a N(u, o²) random variable? 7-17. Suppose U is a Unif(0, 1) random variable, and suppose F is a dis- tribution function and G the corresponding quantile function. Show that the random variable G(U) has F as its DF. Hint: for 0 < u < 1 show that G(u) < x if and only if F(x) > u. This requires use of the fact that DF are right continuous.
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