Stat 116 Practice Final
1. Let
X
1
X
2
X
3
∼
Normal
0
0
1
,
4 1 0
1 1 0
0 0 9
(1) Find
f
X
1

X
2
,X
3
(
X
1

X
2
= 1
, X
3
= 5)
(2) Let
Y
1
=
X
1
+
X
2
,
Y
2
=
X
1
/X
2
. Find the joint p.d.f. of
Y
1
,
Y
2
:
f
Y
1
,Y
2
(
y
1
, y
2
).
2. Suppose the volume of a raindrop (
V
i
) can be modeled as a Gamma distribution with
parameter
p
= 2,
λ
= 16 (i.e.
V
i
∼
Gamma(2,16),
E
(
V
i
) = 2
/
16).
(1) Find the moment generating function (MGF) of the total volume of
N
raindrops
where
N
is a ﬁxed number (i.e. The MGF of
V
= Σ
N
i
=1
V
i
).
(2) Now let
N
be a random variable with a Poisson distribution with parameter
λ
= 100. Find
E
(
V
) and
V ar
(
V
).
(3) What is the moment generating function of
V
when
N
∼
Pois(100) (A random
sum of random variables)?
3. Let
X
and
Y
be independent random variables with
X
∼
Gamma(
r,
1) and
Y
∼
Gamma(
s,
1). Let
Z
1
=
X
+
Y
and
Z
2
=
X/
(
X
+
Y
).
(1) Find the joint distribution of
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 Spring '07
 Staff
 Probability, Probability theory

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