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STAT 333 Assignment 1
Due: Friday, Jan 29 at the beginning of class
1.
Consider 10 independent coin flips having probability
p
of landing heads. We say a
changeover occurs whenever an outcome differs from the one preceding it. For example,
if the results of the flips are H H T H T H H T T T then there are five changeovers.
a.
Find the expected number of changeovers.
b.
Find the variance of the number of changeovers.
c.
Describe how the mean and variance of the number of changeovers behave for
different values of
p
. Provide a brief logical explanation.
(Hint: attach indicator variables to pairs of consecutive flips. X
1
for 1
st
and 2
nd
, X
2
for 2
nd
and 3
rd
, etc. Be careful, the indicators are not all independent.)
2.
Consider a Negative Binomial random variable Y ~ NB(
r
,
p
). Prove that Y is a proper rv
iff
p
> 0 by the following methods:
a.
Express Y as the sum of
r
independent Geometric random variables, and apply a
known result about Geometrics.
b.
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 Winter '08
 Chisholm
 Probability

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