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2. Caroline and Sarah have agreed to meet between 5:30pm and 6:30pm for dinner at Witherspoon street. Let X be Caroline's arrival time (in the unit of minutes) and Y be Sarah's arrival time (in the unit of minutes), relative to 6:00pm. Assume that their arrival times are independent. What is the joint pdf of X and Y, if X and Y are normally distributed as N(0, 10²). If they can wait each other for 10 minutes, what is the probability that they actually meet? What is corr(X+Y,Y)?

2. Caroline and Sarah have agreed to meet between 5:30pm and 6:30pm for dinner at Witherspoon street. Let X be Caroline's arrival time (in the unit of minutes) and Y be Sarah's arrival time (in the unit of minutes), relative to 6:00pm. Assume that their arrival times are independent. What is the joint pdf of X and Y, if X and Y are normally distributed as N(0, 10²). If they can wait each other for 10 minutes, what is the probability that they actually meet? What is corr(X+Y,Y)?

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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**Problem Statement:** Caroline and Sarah have agreed to meet between 5:30 pm and 6:30 pm for dinner at Witherspoon Street. Let \( X \) be Caroline’s arrival time (in units of minutes) and \( Y \) be Sarah’s arrival time (in units of minutes), both relative to 6:00 pm. Assume their arrival times are independent. 1. What is the joint probability density function (pdf) of \( X \) and \( Y \), if \( X \) and \( Y \) are normally distributed as \( N(0, 10^2) \)? If they can wait for each other for 10 minutes, what is the probability that they actually meet? 2. What is \( \text{corr}(X + Y, Y) \)?
Transcribed Image Text:**Problem Statement:** Caroline and Sarah have agreed to meet between 5:30 pm and 6:30 pm for dinner at Witherspoon Street. Let \( X \) be Caroline’s arrival time (in units of minutes) and \( Y \) be Sarah’s arrival time (in units of minutes), both relative to 6:00 pm. Assume their arrival times are independent. 1. What is the joint probability density function (pdf) of \( X \) and \( Y \), if \( X \) and \( Y \) are normally distributed as \( N(0, 10^2) \)? If they can wait for each other for 10 minutes, what is the probability that they actually meet? 2. What is \( \text{corr}(X + Y, Y) \)?
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