ISyE 2027
Probability with Applications
Fall 2011
R. D. Foley
Homework 4
September 15, 2011
due Wednesday
1. Suppose you’re dealt 6 cards from a standard deck. Let
X
be the number
of hearts in the hand. (a) Compute the p.m.f. of
X
. Let
Y
be the number
of clubs in the hand. (b) Compute the joint p.m.f. of
X
and
Y
. That is,
compute
P
{
X
=
i
,
Y
=
j
}
for all
i
and
j
. (Please leave your answer in the
form of a nice expression involving terms like
(
n
k
)
.)
2. Suppose
P
{
X
=
k
}
=
±
ck
2
for
k
=
±
1,
±
2
,
0
otherwise.
(a) What is the correct value of
c
? (b) Compute
P
{
X
= 
1
}
. (c) Compute
E
[
X
]
. (Was that answer surprising?) (d) Compute
E
[
X
2
]
. (e) Find all
medians of
X
. (f) Is the following true:
E
[
X
2
] =
E
[
X
]
2
?
3. Let
A
be an event. Let
I
A
be the indicator random variable of the event
A
. That is,
I
A
=
±
1
if
A
occurs,
0
if
A
does not occur
(a) Compute the p.m.f. of
I
A
. (b) Compute
E
[
I
A
]
. (c) Compute
E
[
I
2
A
]
.
4. Let
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 Fall '08
 Zahrn
 Probability theory

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