Math180AFA22 - Homework 7

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Math 180A Homework 7 Fall 2022 Due date: 11:59pm (Pacific Time) on Wed., Nov. 17 (via Gradescope ) In the "collaborators" field in Gradescope, please write a list of everyone with whom you collaborated on this assignment, as well as any outside sources you consulted, apart from the textbook and your notes. If you did not collaborate with anyone, please explicitly write, "No collaborators." Section 1 (input directly in Gradescope) Submit the answers to these problems directly through the Gradescope interface. You do not need to write up or explain your work. Problem 1. True or false: For all random variables X , Y , and Z , Corr( X + Y, Z ) = Corr( X, Z ) + Corr( Y, Z ) . Problem 2 (numerical answers) . Let X and Y be random variables with E [ X ] = 1 , E [ Y ] = 2 , E [ X 2 ] = 3 , E [ Y 2 ] = 13 , E [ XY ] = 1 . (a) Compute Cov( X, Y ). (b) Compute Corr( X, Y ). (c) Compute Cov( X 1 , 2 Y ). Section 2 (upload files) For each problem, write your solution on a page by itself, and upload it as a separate file to Grade- scope (either typed or scanned from handwritten work). You should write your solutions to these problems neatly and carefully and provide full justification for your answers. Problem 3. Let X be the number of tails in three flips of a fair coin. Let Y be the outcome of a roll of a fair six-sided die. Assume X and Y are independent. Give the joint probability mass function of X, Y . Be precise about the values for which the joint probability mass function is defined. 1
Problem 4. Suppose that X , Y are jointly continuous with joint probability density function f ( x, y ) = ce x 2 2 ( x y ) 2 2 , ( x, y ) R 2 , for some constant c . (a) Find the value of the constant c . (b) Find the marginal density functions of X and Y . (c) Determine whether X and Y are independent. Problem 5. Let X and Y be independent Bernoulli random variables with parameters p and r , respectively. Find the distribution of X + Y . Problem 6. Let X and Y be two independent random variables with densities f X ( x ) = xe x , x 0 , 0 x < 0 and f Y ( y ) = ye y , y 0 , 0 y < 0 Find the PDF of Z = X + Y . Problem 7. Let ( X, Y ) be a uniformly chosen point in the square with corners (1 , 0), (0 , 1), ( 1 , 0), and (0 , 1). (a) What is the joint density function f X,Y : R 2 [0 , )? (b) What is the correlation Corr( X, Y )? (c) Are X and Y independent? Why or why not? 2
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