{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Final-Fall2007-08

# Final-Fall2007-08 - that the ﬁrst to arrive has to wait...

This preview shows page 1. Sign up to view the full content.

American University of Beirut STAT 230 Introduction to Probability and Random Variables Fall 2007-2008 Final Exam Name: ............................... ID #: ................................. Exercise 1 Let X 1 , X 2 , X 3 be three independent random variables with binomial distributions b (4 , 1 / 2) , b (6 , 1 / 3), and b (12 , 1 / 6) respectively. a. find P ( X 1 = 2 , X 2 = 2 , X 3 = 5) b. find E ( X 1 X 2 X 3 ) c. find the mean and the variance of Y = X 1 + X 2 Exercise 2 The joint pdf of two random variables X and Y is f ( x, y ) = kx 0 < x < y < 1 a. find the value of the constant k b. find the marginal pdf of X and Y . Are they independent? c. find P ( X + Y < 1 / 2) d. find E ( X 2 Y ) Exercise 3 A man and a woman decide to meet at a certain location. If each person indepen- dently arrives at a time uniformly distributed between 12 noon and 1 PM, find the probability
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: that the ﬁrst to arrive has to wait no longer than 10 minutes. Exercise 4 Let X and Y be a random sample of from the exponential distribution with pdf f ( x ) = e-x < x < ∞ Let U = X/Y and V = X + Y . Find the joint pdf of the couple ( U,V ). Are U and V independent? Exercise 5 Let X 1 ,X 2 ,...,X n be a random sample of size n with uniform distribution over (0 , 1). Find the pdf of Y = max( X 1 ,X 2 ,...,X n ). Find E ( Y ) ,V ar ( Y ) and M Y ( t ), the moment generating function of Y . good luck...
View Full Document

{[ snackBarMessage ]}