IEOR 4106, Spring 2011, Professor Whitt
Topics for Discussion: Tuesday, April 12
Renewal Theory: Patterns
1. Patterns: see
§
3.6.4 and
§
7.9
Consider successive independent flips of a biased coin.
On each flip, the coin comes up
heads (H) with probability
p
or tails (T) with probability
q
= 1

p
, where 0
< p <
1.
A
given segment of finitely many consecutive outcomes is called a
pattern
. The pattern is said to
occur at flip
n
if the pattern is completed at flip
n
. For example, the pattern
A
≡
HTHTHT
occurs at flips 8 and 10 in the sequence
TTHTHTHTHTTTTHHHT . . .
and at no other
times among the first 17 flips.
WARMUP
For parts (a) and (b) below, assume that
p
= 1
/
2, but for later parts do not make that
assumption.
(a) Which pattern occurs more frequently in the long run:
A
≡
HHH
or
B
≡
HTH
?
(b) For patterns
A
and
B
in part (a), let
N
A
and
N
B
be the numbers of flips until the
patterns
A
and
B
, respectively, first occur. Is
E
[
N
A
] =
E
[
N
B
]?
MAIN PROBLEM
Now we revert to general probabilities
p
and
q
= 1

p
.
(c) What is the probability that pattern
A
≡
HTHTHT
occurs at flip 72?
(d) Suppose that pattern
A
from part (c) does indeed occur at flip 72. What is the expected
number of flips until pattern
A
occurs again?
(e) Let
N
A
(
n
) be the number of occurrences of pattern
A
in the first
n
flips, where
A
is
again the pattern in part (c). Does
N
A
(
n
)
n
→
x
as
n
→ ∞
w.p.1?
If so, what is the limit
x
?
(f) What is
E
[
N
A
], the expected number of flips until pattern
A
≡
HTHTHT
first
occurs?
(g) What is the probability that pattern
A
occurs before pattern
B
≡
TTH
? That is, what
is
P
(
N
A
< N
B
)?
NOTATION
(i)
A
or
B
, a possible pattern, as in parts (a) and (c) above
(ii)
N
A
, a random variable, equal to the number of flips until the pattern
A
occurs for the
first time, as defined in part (b) above
(iii)
A
(
n
), the event that pattern
A
occurs (ends) on flip
n
. Note that in part (c), we are
asking for the value of
P
(
A
(72)).
(iv)
T
A
, a random variable, equal to the number of flips until the pattern
A
occurs again,
after it has occurred previously. This formalized the question in part (d).
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(v)
N
A
(
n
), a random variable, equal to the number of times that pattern
A
occurs in the
first
n
flips, as defined above in part (e)
(vi)
N
A
→
B
, a random variable, equal to the number of flips until the pattern
B
first occurs
starting from the occurrence of
A
. This is new notation used to derive the answers to parts
(f) and (g).
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 Spring '11
 WhitWard
 Probability theory, TA, HT HT

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