STAT 333 Assignment 1
Due: Thursday, May 27 at the beginning of class
1.
Consider rolling a fair 6sided die
n
times. Let X represent the number of faces that have
NOT been rolled.
a.
Find the expected value of X.
b.
Find the variance of X.
c.
Describe (in words or with a graph) how the mean and variance of X behave for
different values of
n
. Provide a brief logical explanation.
(Hint: attach indicator variables to faces, not rolls. Be careful, the indicators are not indep)
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
result we know from class about Geometric rvs.
b.
Show (directly) that P(Y = ∞) = 0.
c.
Show (from first principles – i.e. using the pmf of a NB, or of a Geo) that E[Y] is
r
/
p
. Why does this imply Y is proper iff
p
> 0?
3.
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 Spring '08
 Chisholm
 Variance, Probability theory, fair 6sided die, Geometric random variables, brief logical explanation

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