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Solve each problem. Explain your reasoning. No credit for answers with no
explanation. If the problem is a proof, then you need words as well as formulas.
Explain why your formulas follow one from another.
91.
Suppose
E
(
Y

X
) =
X
var(
Y

X
) = 3
X
2
and suppose the marginal distribution of
X
is
N
(
μ,σ
2
).
(a) Find
E
(
Y
).
(b) Find var(
Y
).
92.
Suppose
X
1
,
...
,
X
N
are IID having mean
μ
and variance
σ
2
where
N
is
a Poi(
λ
) random variable independent of all of the
X
i
. Let
Y
=
N
X
i
=1
X
i
,
with the convention that
N
= 0 implies
Y
= 0.
(a) Find
E
(
Y
).
(b) Find var(
Y
).
93.
Suppose that the conditional distribution of
Y
given
X
is Poi(
X
), and
suppose that the marginal distribution of
X
is Gam(
α,λ
).
Show that the
marginal distribution of
Y
is a negative binomial distribution in the extended
sense discussed in the brand name distributions handout in which the shape
parameter need not be an integer. Identify the parameters of this negative
binomial distribution (which are functions of
α
and
λ
).
94.
Suppose that
X
and
Y
are independent Poisson random variables having
means
μ
1
and
μ
2
, respectively. Show that the conditional distribution of
X
given
X
+
Y
is binomial. Identify the parameters of this binomial distribution
(which are functions of
μ
1
and
μ
2
and
X
+
Y
).
95.
Suppose that
X
has the Geo(
p
) distribution, show that the conditional
distribution of
X

k
given
X
≥
k
also has the Geo(
p
) distribution.
96.
Suppose service times of customers in line at a bank teller are IID Exp(
λ
)
random variables. Suppose when you arrive there are nine customers in line in
front of you (ten customers including you). What is the mean and standard
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 Fall '02
 Staff
 Statistics

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