Assignment6
1. Let
X
and
Y
have joint pdf:
f
X,Y
(
x, y
) =
k
(
x
+
y
)
for
0
≤
x
≤
1
,
0
≤
y
≤
1
.
(a) Find
k
.
(b) Find the joint cdf of (X,Y).
(c) Find the marginal pdf of
X
and of
Y
.
(d) Find
P
[
X < Y
],
P
[
Y < X
2
],
P
[
X
+
Y >
0
.
5].
2. The random vector (
X, Y
) is uniformly distributed (i.e.,
f
(
x, y
) =
k
) in the regions
shown in the following figures and zero elsewhere.
1
1
(
i
)
1
1
(
ii
)
1
1
(
iii
)
(a) Find the value of
k
in each case.
(b) Find the marginal pdf for
X
and for
Y
in each case.
(c) Find
P
[
X >
0
, Y >
0].
3. Let
X
and
Y
be independent random variable. Find the expression for the proba
bility of the following events in terms of
F
X
(
x
) and
F
Y
(
y
).
1
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4. Let
X
and
Y
be the jointly Gaussian random variables with the means
m
1
and
m
2
and variances
σ
1
and
σ
2
respectively. The pdf is as following:
f
X,Y
(
x, y
) =
exp
braceleftBigg

1
2(1

ρ
2
X,Y
)
bracketleftbig(
x

m
1
σ
1
)
2

2
ρ
X,Y
(
x

m
1
σ
1
)(
y

m
2
σ
2
)
+
(
y

m
2
σ
2
)
2
bracketrightbig
bracerightBigg
2
πσ
1
σ
2
radicalBig
1

ρ
2
X,Y
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 Winter '10
 khkjk
 Probability theory, Gaussian random variables

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