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PROBABILITY – Ph.D. Qualifying Exam Spring 2012
(i)
(8 pts)
For each of the following, determine whether the statement is true or false.
(a) If A and B are mutually exclusive, then:
P [A ∪ B] = P [A] + P [B] − P [A]P [B].
(1 pt)
Circle one: (True, False)
EXPLANATION / JUSTIFICATION:
E[X]
X
=
.
(b) If X and Y are independent random variables, then E
Y
E[Y ]
(1 pt)
Circle one: (True, False)
EXPLANATION / JUSTIFICATION:
(c) If E[X|Y ] = E[X] then the variables X and Y are uncorrelated.
Circle one: (True, False)
EXPLANATION / JUSTIFICATION:
(2 pts)
(d) Suppose X and Y can each be either 0 or 1. If X and Y are uncorrelated, then
they are also independent.
(2 pts)
Circle one: (True, False)
EXPLANATION / JUSTIFICATION:
(e) If X1 and X2 are independent, identically distributed, zero-mean Gaussian random variables with variance of 1, then Z = (X1 + X2 )/2 is also a zero-mean
Gaussian random variable with variance of 1.
(2 pts)
Circle one: (True, False)
EXPLANATION / JUSTIFICATION:
(ii)
(6 pts)
For each of the cases shown in (a). . . (d) below, determine whether the random variables X and Y are independent. If they are independent, then calculate and sketch
the marginal PDFs fX (x) and fY (y). In all cases, you may assume that the joint
PDF is zero at all points outside of the unit square.
y
y
1
y
1
y
1
1
fXY(x,y) = 4x
fXY(x,y) = 1
fXY(x,y) = 1
fXY(x,y) = 2x
fXY(x,y) = 1
1/2
fXY(x,y) = 2x
1
(a)
x
1
(b)
x
1
(c)
x
1/2
(d)
1
x
(iii)
(6 pts)
Suppose X is an exponential random variable characterized by the distribution
(
e−x , x ≥ 0
fX (x) =
0,
otherwise
and Y is related to X by Y = X 2 . Find the probability density function for Y .