## Contents

## Instructions

Homework 4 can be downloaded here on the ECE 302 course website.

## Problem 1: Binomial Proofs

Let $ X $ denote a binomial random variable with parameters $ (N, p) $.

- (a) Show that $ Y = N - X $ is a binomial random variable with parameters $ (N,1-p) $

4.1a Suan-Aik Yeo_ECE302Fall2008sanghavi

4.1a Emir Kavurmacioglu_ECE302Fall2008sanghavi

4.1a Eric Zarowny_ECE302Fall2008sanghavi comment by Beau Morrison

4.1a Jared McNealis_ECE302Fall2008sanghavi

4.1a Virgil Hsieh_ECE302Fall2008sanghavi

- (b) What is $ P\{X $ is even}? Hint: Use the binomial theorem to write an expression for $ (x + y)^n + (x - y)^n $ and then set $ x = 1-p $, $ y = p $.

4.1b Steve Streeter_ECE302Fall2008sanghavi

4.1b Steven Millies_ECE302Fall2008sanghavi

4.1b Junzhe Geng_ECE302Fall2008sanghavi

## Problem 2: Locked Doors

An absent-minded professor has $ n $ keys in his pocket of which only one (he does not remember which one) fits his office door. He picks a key at random and tries it on his door. If that does not work, he picks a key again to try, and so on until the door unlocks. Let $ X $ denote the number of keys that he tries. Find $ E[X] $ in the following two cases.

- (a) A key that does not work is put back in his pocket so that when he picks another key, all $ n $ keys are equally likely to be picked (sampling with replacement).

4.2a Ken Pesyna_ECE302Fall2008sanghavi

4.2a Spencer Mitchell_ECE302Fall2008sanghavi

4.2a Hamad Al Shehhi_ECE302Fall2008sanghavi

4.2a Joe Romine_ECE302Fall2008sanghavi

4.2a Ben Carter_ECE302Fall2008sanghavi in response to Joe's Comment

4.2a Christopher Wacnik_ECE302Fall2008sanghavi

4.2a Phil Cannon_ECE302Fall2008sanghavi

- (b) A key that does not work is put in his briefcase so that when he picks another key, he picks at random from those remaining in his pocket (sampling without replacement).

4.2b Brian Thomas_ECE302Fall2008sanghavi

4.2b AJ Hartnett_ECE302Fall2008sanghavi

4.2 Arie Lyles_ECE302Fall2008sanghavi question about classifying random variables

4.2b Gregory Pajot_ECE302Fall2008sanghavi Note about arithmetic series, and random variable classification

4.2b Henry Michl_ECE302Fall2008sanghavi More general sum of arithmetic series explanation

## Problem 3: It Pays to Study

There are $ n $ multiple-choice questions in an exam, each with 5 choices. The student knows the correct answer to $ k $ of them, and for the remaining $ n-k $ guesses one of the 5 randomly. Let $ C $ be the number of correct answers, and $ W $ be the number of wrong answers.

- (a) What is the distribution of $ W $? Is $ W $ one of the common random variables we have seen in class?
- (b) What is the distribution of $ C $? What is its mean, $ E[C] $?

4.3 Tiffany Sukwanto_ECE302Fall2008sanghavi

4.3a Priyanka Savkar_ECE302Fall2008sanghavi

4.3 Monsu Mathew_ECE302Fall2008sanghavi

4.3 Zhongtian Wang_ECE302Fall2008sanghavi-comments for Monsu Mathew

4.3b Hamad Al Shehhi_ECE302Fall2008sanghavi- The E[C]

4.3b Ashton Frierson_ECE302Fall2008sanghavi - The E[C]

4.3b Steve Anderson_ECE302Fall2008sanghavi - Comment regarding previous comments about 4.3b

4.3a Sourabh Ranka_ECE302Fall2008sanghavi

4.3 Carlos Leon_ECE302Fall2008sanghavi

4.3 Joon Young Kim_ECE302Fall2008sanghavi - i still have a trouble to understand about the meaning of common random variable.

4.3b Andrew Hermann_ECE302Fall2008sanghavi - when to add k

## Problem 4: No Deal

In "Deal or No Deal" (the most ridiculous game show on TV), there are 5 suitcases. The suitcases contain $1, $10, $100, $1,000 and $10,000, respectively. There is a "banker" who offers the contestant a dollar amount that he can take and go home, right then and there. If the contestant does not use the banker's offer, he can choose one of the suitcases and "eliminate" it by removing it from play. Then he plays the next round with the remaining suitcases.

- (a) The banker wants to offer an amount equal to the average of what will REMAIN, after the choice is made. (for example, if 1000 is chosen, then

the average of what will remain is (1 + 10 + 100 + 10000)/4.) Of course, the banker has to make an offer before the choice is made. What amount should the banker offer?

- (b) The contestant has nerves of steel, and never takes up the banker's offer in any round. He thus goes home with one of the 5 suitcases. However, he has to pay a 30% tax on the amount he takes home. How much will he be left with on average, after taxes?

4.4 Zhongtian Wang & Jonathan Morales_ECE302Fall2008sanghavi

4.4 Anand Gautam_ECE302Fall2008sanghavi

4.4A Katie Pekkarinen_ECE302Fall2008sanghavi

4.4B Shao-Fu Shih_ECE302Fall2008sanghavi

4.4b Michael Allen_ECE302Fall2008sanghavi

4.4b Seraj Dosenbach_ECE302Fall2008sanghavi

4.4b Joe Gutierrez_ECE302Fall2008sanghavi

4.4b Ben Wurtz_ECE302Fall2008sanghavi

4.4 Joshua Long_ECE302Fall2008sanghavi

4.4b Shweta Saxena_ECE302Fall2008sanghavi

4.4b Jayanth Athreya_ECE302Fall2008sanghavi