2027A8

ISyE 2027: Probability with Applications Handout 18 Sigr´un Andrad´ottir October 28, 2021 Assignment 8 Due November 4, 2021, 11 PM Remember to attach a fully completed cover sheet to your homework Problem 1 (6 points) Do Problem 4 on Assignment 7. Problem 2 (17 points) a. (2 points) Consider discrete random variables X and Y whose joint probability mass function (PMF) p X,Y ( a, b ) is provided in the following table: p X,Y ( a, b ) b = 5 b = 10 a = 1 0 . 15 p a = 2 0 . 45 0 . 05 ( p X,Y ( a, b ) = 0 if a 6∈ { 1 , 2 } or b 6∈ { 5 , 10 } ). Determine the value of p (if possible). Explain. b. (2 points) Determine the joint cumulative distribution function (CDF) F X,Y of the random variables X and Y defined in part a . Make sure to fully specify this function. Explain. c. (5 points) Do Problem 9.1, page 127 in the text. Also, compute the marginal cumulative distribution functions (CDFs) of X and Y . Make sure to fully specify the functions. Are X and Y independent? Explain. d. (4 points) Do Problem 9.3, page 129 in the text (recall that { 1 , 4 } × { 1 , 4 } = { (1 , 1) , (1 , 4) , (4 , 1) , (4 , 4) } ). Explain. e. (4 points) Suppose 0 < p, q, r < 1 and consider the random variables Z 1 , Z 2 , and Z 3 defined as follows: Z 1 = ( 1 with probability p, 2 with probability 1 - p ; Z 2 = ( 5 with probability q, 10 with probability 1 - q ; Z 3 = ( 10 with probability r, 20 with probability 1 - r. Assume that the random variables Z 1 , Z 2 , and Z 3 are independent. Define the random variables X = Z 1 + Z 2 and Y = Z 1 + Z 3 . Specify the joint probability mass function (PMF) of X and Y . Make sure to fully specify this function. Are X and Y independent? Explain. 1

Problem 3 (6 points) a. (2 points) Consider the function: f ( x, y ) = ( αx if 1 ≤ x ≤ 2, 2 ≤ y ≤ 4 , 0 otherwise . Is there a value of α for which f is the joint probability density function (PDF) of a pair of random variables X and Y ? If so, then specify that value of α . Explain. b. (4 points) Consider random variables X and Y with joint probability density func- tion (PDF): f X,Y ( x, y ) = ( 72 7 x 2 y ( x + y ) if 0 ≤ x ≤ y ≤ 1 , 0 otherwise . Determine P (0 . 2 ≤ X ≤ 0 . 8 , 0 . 4 ≤ Y ≤ 0 . 6) and P ( X ≤ Y 2 ). Explain. Problem 4 (8 points) Consider random variables X and Y with joint probability density function (PDF): f X,Y ( x, y ) = ( 72 17 x 2 y ( x + y ) if 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 , 0 otherwise . a. (4 points) Compute the marginal probability density functions (PDFs) f X and f Y and the marginal cumulative distribution functions (CDFs) F X and F Y of the random variables X and Y . Make sure to fully specify these functions. Explain. b. (2 points) Determine the joint cumulative distribution function (CDF) F X,Y of the random variables X and Y . Make sure to fully specify this function. Explain. c. (2 points) Are the random variables X and Y independent? Explain. Problem 5 (7 points) Consider random variables X and Y with joint cumulative distribution function (CDF): F X,Y ( a, b ) =        0 if a < 0 or b < 0 , a 2 (1 - e - 2 b ) 4 if 0 ≤ a ≤ 2 and b ≥ 0 , 1 - e - 2 b if a > 2 and b ≥ 0 . a. (2 points) Compute the marginal cumulative distribution functions (CDFs) F X and F Y of the random variables X and Y . Make sure to fully specify these functions. Explain. b. (2 points) Determine the joint probability density function (PDF) f X,Y of the ran- dom variables X and Y . Make sure to fully specify this function. Explain. c. (3 points) Let Z 1 = √ X and Z 2 = Y 2 . Are the random variables Z 1 and Z 2 independent? Explain. Reading Assignment Read Chapter 9 of the text. 2