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Problem 6. The hat-check staff has had a long day, and at the end of the party they decide to return people's hats at random. Suppose that n people have their hats returned at random. We previously showed that the expected number of people who get their own hat back is 1, irrespective of the total number of people. In this problem we will calculate the variance in the number of people who get their hat back. Let X; = 1 if the ith person gets his or her own hat back and 0 otherwise. Let S, := E, Xi, so S, is the total number of people who get their own hat back. Show that vi=1 (a) E[X?] = 1/n. %3| (b) E [X;X;] = 1/n(n – 1) for i + j. (c) E[S] = 2. Hint: Use (a) and (b). (d) Var [S„] = 1. %3D

Problem 6. The hat-check staff has had a long day, and at the end of the party they decide to return people's hats at random. Suppose that n people have their hats returned at random. We previously showed that the expected number of people who get their own hat back is 1, irrespective of the total number of people. In this problem we will calculate the variance in the number of people who get their hat back. Let X; = 1 if the ith person gets his or her own hat back and 0 otherwise. Let S, := E, Xi, so S, is the total number of people who get their own hat back. Show that vi=1 (a) E[X?] = 1/n. %3| (b) E [X;X;] = 1/n(n – 1) for i + j. (c) E[S] = 2. Hint: Use (a) and (b). (d) Var [S„] = 1. %3D

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 6. The hat-check staff has had a long day, and at the end of the party they decide to return people's hats at random. Suppose that n people have their hats returned at random. We previously showed that the expected number of people who get their own hat back is 1, irrespective of the total number of people. In this problem we will calculate the variance in the number of people who get their hat back. Let X; = 1 if the ith person gets his or her own hat back and 0 otherwise. Let S, := E X;, so S, is the total number of people who get their own hat back. Show that (a) E[X?] = 1/n. (b) E[X;X;] = 1/n(n – 1) for i + j. (c) E [S2] = 2. Hint: Use (a) and (b). (d) Var [S„] = 1.
Transcribed Image Text:Problem 6. The hat-check staff has had a long day, and at the end of the party they decide to return people's hats at random. Suppose that n people have their hats returned at random. We previously showed that the expected number of people who get their own hat back is 1, irrespective of the total number of people. In this problem we will calculate the variance in the number of people who get their hat back. Let X; = 1 if the ith person gets his or her own hat back and 0 otherwise. Let S, := E X;, so S, is the total number of people who get their own hat back. Show that (a) E[X?] = 1/n. (b) E[X;X;] = 1/n(n – 1) for i + j. (c) E [S2] = 2. Hint: Use (a) and (b). (d) Var [S„] = 1.
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