# Ps5

.pdf
School
Arizona State University **We aren't endorsed by this school
Course
STP 421
Subject
Statistics
Date
Aug 31, 2023
Pages
3
STP 421 - Spring 2023 Problem Set 5 Due: April 28 2023 Instructions: Show your work and attempt every problem! Partial credit will be given if you can show some progress towards a solution. If you can't solve a problem in full generality, try to solve some special cases. You may work in groups and consult outside resources (textbooks, web sites, etc.). How- ever, you should acknowledge any assistance received from other people or outside re- sources. If working in a group, please list the names of your group members. Submitted solutions should either be typed, preferably using LaTeX, or neatly handwrit- ten. Solutions should be submitted as PDF documents through Canvas. If submitting scanned images, please assemble all pages into a single PDF document prior to uploading to Canvas. 1
1.) Suppose that X and Y are jointly continuous random variables with joint density function f ( x, y ) = c ( y 2 - x 2 ) e - 2 y , - y x y, 0 < y < . (a) Find c so that f is a density function. (b) Find the marginal densities of X and Y . (c) Find the expected value of X . 2.) Suppose that X and Y are jointly continuous random variables with joint density function f ( x, y ) = xe - x ( y +1) , 0 x, y < . (a) Find the conditional densities of X and Y . (b) Find the cumulative distribution function and density of Z = XY . 3.) Suppose that X 1 , · · · , X n are independent normal random variables with distributions N ( μ 1 , σ 2 1 ) , · · · , N ( μ n , σ 2 n ), respectively, and let X = c 1 X 1 + · · · + c n X n , where c 1 , · · · , c n are real numbers. Find the distribution of X . 4.) Let Z 1 , Z 2 be independent exponential random variables each with parameter 1. Find the cumulative distribution function of the random variable X = Z 1 Z 1 + Z 2 and identify the distribution of X . 5.) Suppose that X 1 , · · · , X n are independent, identically-distributed random variables with marginal density function f ( x ). Calculate the probability P ( X 1 < X 2 < · · · < X n ) . 6.) For each n 1, let X n be a Poisson-distributed random variable with parameter λ n = n and define Y n = n - 1 / 2 ( X n - n ) . Show that the sequence ( Y n ; n 1) converges in distribution to a standard normal random variable. 2
7.) Let X be a standard normal random variable, let I be independent of X with distribution P { I = +1 } = P { I = - 1 } = 1 / 2, and let Y = I · X . (a) Show that Y is also a standard normal random variable. (b) Show that Cov( X, Y ) = 0. (c) Show that X and Y are not independent. 3
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