12/5/03 3:25 PM
HW14 Solution page 1
Probability for Engineering Applications
SOLUTION Assignment #14
1. (1 pt)
Express VAR[X+Y] in terms of E[X], E[Y], VAR[X], VAR[Y], and E[XY].
What happens if E[XY]= E[X] E[Y] ? (cf. 4.59)
VAR[X+Y]
= E[(X+Y)
2
] – (E[X+Y])
2
=
E[X
2
] +2E[XY] + E[Y
2
] – (E[X])
2
– 2E[X]E[Y] – (E[Y])
2
= VAR[X] + VAR[Y] +2E[XY] – 2E[X]E[Y]
VAR[X+Y] = VAR[X] +VAR[Y] only when X and Y are independent
(i.e., when E[XY]=E[X]E[Y])
2. (1 pt)
Find E[X
2
Y] where X is a zero-mean, unit-variance Gaussian variable, and Y is
a uniform random variable in [-2, +5], and X and Y are independent.
X and Y are independent
]
[
]
[
]
[
2
2
Y
E
X
E
Y
X
E
=
⇒
X is zero-mean, unit-variance Gaussian r. v.
1
]
[
=
⇒
X
VAR
,
0
]
[
=
X
E
1
])
[
(
]
[
]
[
2
2
=
+
=
X
E
X
VAR
X
E
5
.
1
2
))
2
(
5
(
]
[
=
-
+
=
Y
E
5
.
1
5
.
1
1
]
[
]
[
]
[
2
2
=
×
=
=
Y
E
X
E
Y
X
E
3. (2pts) Random variables X and Y have the pmf shown below.
(This is the same pmf as in HW 11) Are the variables independent? Correlated?
What is
ρ
X,Y
? (cf. 4.64)
Y
-1
0
+1
-1
0.10
0.05
0.10
0
0.05
0.15
0.15
X
+1
0.25
0.10
0.05
Marginal PMF:
25
.
0
10
.
0
05
.
0
10
.
0
]
1
[
35
.
0
15
.
0
15
.
0
05
.
0
]
0
[
4
.
0
05
.
0
10
.
0
25
.
0
]
1
[
=
+
+
=
-
=
=
+
+
=
=
=
+
+
=
=
X
P
X
P
X
P
4
.
0
25
.
0
05
.
0
10
.
0
]
1
[
3
.
0
10
.
0
15
.
0
05
.
0
]
0
[
3
.
0
05
.
0
15
.
0
10
.
0
]
1
[
=
+
+
=
-
=
=
+
+
=
=
=
+
+
=
=
Y
P
Y
P
Y
P
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