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C
OMER
ECE 600 Homework 5
1.
The hat check staff has had a long day, and at the end of a party they decide to return
people’s hats at random. Suppose that
n
people have their hat returned at random.
a.
What is the expected number of people who get the correct hat back?
b.
In the limit as
n
±
, what is the answer to part (a). Do you expect anyone to get
his/her hat back?
2.
There are
n
bags numbered 1,…,
n
and an infinite supply of balls. You pick up a ball and it is
equally likely that you put it in any of the
n
bags.
a.
What is the expected number of balls you need to put in the
n
bags before the
i
th
bag
has at least one ball?
b.
Show that the expected number of balls that you need to put in before every bag has
at least one ball is
n
H(
n
), where
1
1
1
H( )
1
2
1
n
n
n
=
+
+
+
+

3.
Let the joint cumulative distribution function (cdf) of random variables X and Y be F
XY
(
x,y
).
Show that
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 Spring '08
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