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).
