Download Stat 430 Homework 2 Fall 2008

Stat 430
Homework 2
Fall 2008
Maximum score is 24 points, due date is Tuesday, Sep 6th. You can either hand in the solution electronically
or on paper during class.
1
Birthday Problem
(a) Consider a class of 30 students. What is the probability that at least two of them share the same
birthday? (same day, not the same year; ignore leap years)
(b) How many students should be at least in the class to have this probability above 0.5?
(4 points)
2
Password
A computer user tries to recall her password. She knows it can be one of four possible passwords. She tries
her passwords until she finds the right one. Let X be the number of wrong passwords she uses before she
finds the right one. Determine an appropriate sample space, and find the probability mass function for X.
Determine E[X] and V ar[X].
(4 points)
3
Network Blackouts
Every day the number of computer blackouts has a distribution:
0
x
p(x) 0.7
1
0.2
2
0.1
A small internet trading company estimates that each network blackout results in a $500 loss. Compute
expected value and variance of this company’s daily loss due to blackouts.
(2 points)
4
Joint Distribution
Two random variables X and Y have the joint distribution P (x, y)
P (x, y)
y
0
1
x
0
0.5
0.2
1
0.2
0.1
(a) Are X and Y independent? Explain.
(b) Are (X + Y ) and (X − Y ) independent? Explain.
(3 points)
5
Hardware Failures
The number of hardware failures X and the number of software failures Y on any day in a small computer
lab have the joint distribution P (x, y) where P (0, 0) = 0.6, P (1, 0) = 0.1, P (0, 1) = 0.1, and P (1, 1) = 0.2.
Based on this information,
(a) are X and Y independent?
(b) compute E[X + Y ], the expected total number of failures on any given day.
(3 points)
6
Tossing Coins
Two fair coins are tossed and you are told that at least one coin shows a head. What is the probability that
both coins show heads? Why is neither 1/4 nor 1/2 the correct answer?
(2 points)
7
Moment Generating Functions
Find the moment generating functions of the Binomial, the Poisson and the Uniform distribution, respectively.
(6 points)
2