C
OMER
ECE 600 Homework 7
1.
Let X and Y be two independent binomial random variables with pmf
X
Y
(1
)
0
p ( )
p ( )
0
otherwise
k
n
k
n
p
p
k
n
k
k
k

°±
²

≤
≤
³
´
µ
=
=
¶·
¸
³
¹
Let Z = X + Y. Find the pmf of Z. Do you find anything special about the pmf of Z?
Comment on it.
2.
The hatcheck staff has had a long day, and at the end of the party they decide to return
people’s hats at random. Suppose that
n
people have their hats returned at random. You
have previously shown that the expected number of people who get their own hat back is 1,
irrespective of the total number of people. In this problem you are asked to find the variance
of the number of people who get their own hat back. Let X
i
= 1 if person
i
gets his or her
own hat back and 0 otherwise. Let
1
S
X
n
n
i
i
=
=
º
be the total number of people who get their
own hat back.
a.
Show that Var(S
n
) = 1.
b.
Explain why you cannot use the variance of sums formula to calculate Var(S
n
).
3.
Let us consider four zeromean, unitvariance random variables T, U, V, and W. Assume
that they are pairwiseuncorrelated. We next define new random variables X, Y, and Z as X
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