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) $
- (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).
- (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).
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] $?
