STAT 333 Assignment 2
Due: Friday, March 2 at the beginning of class (or up to 10 minutes in)
1.
Consider a sequence of independent tosses of a fair coin. Each toss results in a head H or a tail T.
Let
λ
be the event that the total number of H equals exactly onethird the number of tosses, i.e.
we say
λ
occurs on the n
th
toss if the total number of H up to and including the n
th
toss equals n/3.
a.
Give an argument that
λ
is a renewal event
b.
What is the period of
λ
?
c.
Prove that lambda is transient. Why does this make logical sense when you consider the law
of large numbers? (that with a large number of trials, the percentage of successes will
approach the probability of a success)
2.
Suppose a sequence of independent trials X
1
,X
2
, . . . is generated by randomly picking digits
(with replacement) from the set {0, 1, 2, . . ., 9}. Prove that the event “2 3 5” is recurrent by:
a.
Finding r
n
= P(“2 3 5” occurs on trial n) and showing that Σr
n
= ∞
b.
Obtaining the generating function R
235
(s) of the renewal sequence {r
n
}, finding the pgf
F
235
(s) of the first waiting time T
235
and evaluating F
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 Winter '08
 Chisholm
 Logic, Recurrence relation, delayed renewal

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