homework5 - COMER ECE 600 Homework 5 1 The hat check staff...

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C OMER ECE 600 Homework 5 1. The hat check staff has had a long day, and at the end of a party they decide to return people’s hats at random. Suppose that n people have their hat returned at random. a. What is the expected number of people who get the correct hat back? b. In the limit as n ° ± , what is the answer to part (a). Do you expect anyone to get his/her hat back? 2. There are n bags numbered 1,…, n and an infinite supply of balls. You pick up a ball and it is equally likely that you put it in any of the n bags. a. What is the expected number of balls you need to put in the n bags before the i th bag has at least one ball? b. Show that the expected number of balls that you need to put in before every bag has at least one ball is n H( n ), where 1 1 1 H( ) 1 2 1 n n n = + + + + - ° 3. Let the joint cumulative distribution function (cdf) of random variables X and Y be F XY ( x,y ). Show that ) , ( F ) , ( F ) , ( F ) , ( F }) Y { } X P({ 1 1 XY 1 2 XY 2 1 XY 2 2 XY 2 1 2 1 y x y x y x y x y y x x + - - = < < 4. Determine the constant b
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