STAT 36217, HW 5, due Thursday 10/8/2009, 10:30 AM
1. The joint PMF of
X
and
Y
are given by
p
(1
,
1) = 1
/
9
, p
(2
,
1) = 1
/
3
, p
(3
,
1) =
1
/
9
, p
(1
,
2) = 1
/
9
, p
(2
,
2) = 0
, p
(3
,
2) = 1
/
18
, p
(1
,
3) = 0
, p
(2
,
3) = 1
/
6
, p
(3
,
3) = 1
/
9.
Calculate
E
[
X

Y
=
i
] for
i
= 1
,
2
,
3. Are
X
and
Y
independent?
2. If
X
and
Y
are independent Poisson distribution having the same parameter 1. Cal
culate the conditional distribution of
X
=
k
given
X
+
Y
=
n
.
3. Let
X
1
and
X
2
be independent geometric random variables having the same parameter
p
. Calculate
P
(
X
1
=
i

X
1
+
X
2
=
n
). (Hint: you can first guess the result and the
verify the answer. The idea is a follows. Suppose a coin having probability
p
of coming
up heads is continuously flipped. If a second hear occurs on flip number
n
, what is the
conditional probability that the first head was on flip number
i
,
i
= 1
,
2
, . . . , n

1?).
4. There are
n
components. On a rainy day, component
i
will function with probability
p
i
; on a non rainy day, component
i
will function with probability
q
i
, for
i
= 1
,
2
, . . . , n
.
It will rain tomorrow with probability
α
.
Calculate the conditional expected num
ber of components that function tomorrow, given that it rains. Also, calculate the
expected number of components that function tomorrow.
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 Fall '09
 Jin
 Probability theory, probability density function

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