Korea University
School of Information and Communications
CNCE363: Probability and Random Process
Spring Semester, 2011
Final exam
June 17 Friday (120 minutes)
This is closed book test. However, two A4 pages of cheating sheet are allowed.
Problem 1)[20pt] Let X1and X2be the independent and identically distributed exponential
random variables with the parameter λ(i.e., fZ(z) = λe−λz for z≥0).
(a) Find out P(min(X1, X2)≤a).
(b) Find out P(max(X1, X2)≤a).
Problem 2)[20pt] Let X,Y, and Zbe independent and uniformly distributed on (0,1).
Find P(X > Y Z ).
Problem 3)[20pt] Three points X1,X2, and X3are selected at random on a line with length
L. What is the probability that X2lies between X1and X3?
Hint: You should consider two scenarios; X1< X2< X3and X3< X2< X1.
Problem 4)[20pt] Consider continuous random variables Xand Y. The conditional distribution
of xgiven yis fX|Y(x|y) = ye−xy for 0 < x < ∞and the distribution of yis fY(y) = 3y2for
0< y < 1.
(a) Find E[X|Y=y].
(b) Find E[X]. (Hint: use fundamental theorem of expectation)
Problem 5)[20pt] Moment generating function (MGF) of a RV Xis defined as M(t) = E[etX].
(a) What is M(t) for the standard normal distribution?
(b) Find out E[X] of the standard normal distribution using MGF.
(c) Find out V ar(X) of the standard normal distribution using MGF.