Download EE571m1 F03-04

DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING
EE 571 MIDTERM EXAM I
DATE : 24 Dec. 2003
DURATION : 2 HRS.
1-) There are 4 red balls and 6 white balls in a box.
(a) 10 balls are drawn from the box successively, each one replaced (returned to the
box) after drawing.
(i)
Find the probability that the number of red balls drawn is 4.
(ii)
Find the probability that the number of white balls drawn is 8.
(iii)
Find the probability that the first 6 balls drawn are red.
(b) Two balls are drawn successively and without replacement. Find
(i) The probability that one is red and one is white.
(ii) The probability that the second ball drawn is white.
2-) X is a random variable uniformly distributed in [0,π].
(a) Find and sketch the probability density function (pdf) of Y= sin2X.
(b) Find the probability P{ 0.8 < Y < 1 }.
3-) X and Y are random
D : ( x, y) x 2  y 2  1, y  0 .


variables
jointly
uniform
in
the
semi-disc
(a) Find the marginal pdfs of X and Y ( fX(x) and fY(y) ).
(b) Find E{Y} and E{XY}.
(c) Find the covariance of X and Y (σXY).
4-) X1 and X2 are independent and identically distributed random variables with probability
mass function (pmf)
P{ X  k}  p.q k
; q  1  p ; k  0,1, 2,.....
(a) Find the pmf of Y= X1+ X2 .
(b) Find P{Y  N } in terms of N in closed form ( no summations ).
5-) Find the pdf of Z = X + Y , where X is an exponential random variable with pdf
 e  x
f X ( x)  
0
and Y is uniformly distributed in [0,1].
x0
x0