Korea University
School of Information and Communications
CNCE363: Probability and Random Process
Spring Semester, 2012
Final exam (Closed book and open note)
June 19 Tuesday (in class)
Problem 1)[20pt] Let Xbe the number of 1’s and Ybe the number of 2’s that occur in nrolls
of a fair die. Compute Cov(X, Y ).
Problem 2)[20pt] The continuous random variables Xand Yhave joint distribution fX,Y (x, y) =
1
xfor 0 <y<x<1.
(a) Compute E[X] and E[Y].
(b) Compute Cov(X, Y ).
Problem 3)[20pt] A test engineer discovered that the CDF of the lifetime of an equipment in
years is given by
FX(x) = 1 −e−x/5if 0 <x<∞(1)
(a) What is the expected lifetime (i.e., expected value) of the equipment?
(b) What is the variance of the lifetime of the equipment?
Problem 4)[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).
