## 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 Eric Zarowny_ECE302Fall2008sanghavi comment by Beau Morrison

• (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 $x = 1-p$, $y = p$.

## 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 Ben Carter_ECE302Fall2008sanghavi in response to Joe's Comment

• (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).

4.2 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 Zhongtian Wang_ECE302Fall2008sanghavi-comments for Monsu Mathew

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?

ECE462 Survivor

Seraj Dosenbach