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i) Using the fact that e-* dx = Vn, explain how we can verify that the density function for the normal distribution N 0, () is a valid density function. ii) Show that if X N0, then E(X) = 0. iii) Given that (° x²e-x*dx = ¥ , show that var(X) = }. Let f: R2 → R be given by: 2(1-p2) f(x,y) = 2n (1-p²) for -1

i) Using the fact that e-* dx = Vn, explain how we can verify that the density function for the normal distribution N 0, () is a valid density function. ii) Show that if X N0, then E(X) = 0. iii) Given that (° x²e-x*dx = ¥ , show that var(X) = }. Let f: R2 → R be given by: 2(1-p2) f(x,y) = 2n (1-p²) for -1

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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i) Using the fact that e-x dx = \T , explain how we can verify that the density function for the normal distribution N|0, is a valid density function. ii) Show that if X N 0,(E then E(X) = 0. iii) Given that x²e**dx = , show that var(X) =}. 2 Let f: R2 → R be given by: of f (x, y) = 21 p5(x²-2pxy+y?) 2n (1 - p2) for -1 < p < 1. This function is the joint distribution function for two normally distributed variables X, Y ~ N(0,1), withp = cov(X,Y). iv) Explain why, in this instance, the correlation of X and Y,p(X, Y), is equal to the covariance of X and Y, cov(X,Y). v) Show that if p = 0, then X and Y are independent.
Transcribed Image Text:i) Using the fact that e-x dx = \T , explain how we can verify that the density function for the normal distribution N|0, is a valid density function. ii) Show that if X N 0,(E then E(X) = 0. iii) Given that x²e**dx = , show that var(X) =}. 2 Let f: R2 → R be given by: of f (x, y) = 21 p5(x²-2pxy+y?) 2n (1 - p2) for -1 < p < 1. This function is the joint distribution function for two normally distributed variables X, Y ~ N(0,1), withp = cov(X,Y). iv) Explain why, in this instance, the correlation of X and Y,p(X, Y), is equal to the covariance of X and Y, cov(X,Y). v) Show that if p = 0, then X and Y are independent.
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