Staying with the “coin flipping game”, let 'Z 'record the first point in time when some player (either the “Head” player or the “Tail” player) has a lead of $1, and let 'W' record the first point in time when some player has a lead of $2. Find the least numbers n1 and n2 such that, Prob(Z ≤ n1) > 1/2 and Prob(W ≤ n2) > 1/2. What is E(Z)? What is Prob(W = 2n + 2|W > 2n)? What is Prob(W > 2n)? What is Prob(W = 2n)?
Staying with the “coin flipping game”, let 'Z 'record the first point in time when some player (either the “Head” player or the “Tail” player) has a lead of $1, and let 'W' record the first point in time when some player has a lead of $2. Find the least numbers n1 and n2 such that, Prob(Z ≤ n1) > 1/2 and Prob(W ≤ n2) > 1/2. What is E(Z)? What is Prob(W = 2n + 2|W > 2n)? What is Prob(W > 2n)? What is Prob(W = 2n)?
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1) Staying with the “coin flipping game”, let 'Z 'record the first point in time when some player (either the “Head” player or the “Tail” player) has a lead of $1, and let 'W' record the first point in time when some player has a lead of $2. Find the least numbers n1 and n2 such that, Prob(Z ≤ n1) > 1/2 and Prob(W ≤ n2) > 1/2. What is E(Z)? What is Prob(W = 2n + 2|W > 2n)? What is Prob(W > 2n)? What is Prob(W = 2n)?
* PLEASE READ THE PAGE EXPLANING THE COIN FLIPPING GAME !
![A Random Walk on the Integers: We start with the coin toss game which uses a coin
with probability P for Head and probability q
up Head and lose a dollar it it shows Tail. The coin will be tossed infinitely many times and
the random variable X; measures your earnings from the i'h toss:
1-P for Tail. You get a dollar if it comes
1 if the ith toss is Head
-1 otherwise
X;
Your profit (excess of heads over tails) after m tosses is
Sm = X1+
+ Xm:
Your progress in this game may be thought of as describing a random walk on the integers.
One way to think of this is on the line, or in one dimension: Let So = 0 signify the start at
the origin. Each toss is a step in which "Head" moves you one integer to the right from the
current position and "Tail" moves one integer to the left. Thus S; is the position of the walk
after the jth step (toss). The study of this interesting game reveals some surprising properties,
and it is another very significant example showing the power of generating functions.
• Positive Paths
two dimensions, i.e., in the plane: we use the x-axis for "time" and the y-axis for the
progress of the walk. The sequence of points {(0,0), (1, S1), (2, S2) . , (m, Sm)} describes
the start at (0, 0 [at time zero the game is even]; after step j our point has x-coordinate
j and y-coordinate S; = X1 + .… + X;. (S; is the the excess of heads over tails for the
first j tosses.) If we connect successive points by straight line segments we get what we
will call a random walk path with m steps (starting at the origin, ending at (m, Sm)).
There are 2m such paths (because each sequence of m tosses gives a distinct path). For
now we will take P = 1/2, so each of the 2m paths is equally likely.
The Ballot Theorem Another way to depict the above walk is in
•..)
We write
m = NH + NT
for the number of tosses, nH denoting the number of Heads and nT the number of Tails,
and observe that the score (our fortune) after m tosses is the difference between the
number of Heads and the number of Tails, or
— Пн — Пт:
Sm
Fix a positive integer k, 0 < k < m, and let us focus our attention on the paths that go
from (0, 0) to (m, k); i.e. Sm = k is the score after m steps.
Observation 1: Since Sm = NH
6.
NT
2nH – m, k is even if m is, and vice-versa.
Also note that the number of distinct paths which start at (0, 0) and end at (m, k) is
m
Nm,k](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6971cef0-9a3b-4ce4-80d1-ccd4d229c95d%2F96ba7490-4231-496a-b1f6-ddfe056fe0ca%2Fpte1zbq_processed.png&w=3840&q=75)
Transcribed Image Text:A Random Walk on the Integers: We start with the coin toss game which uses a coin
with probability P for Head and probability q
up Head and lose a dollar it it shows Tail. The coin will be tossed infinitely many times and
the random variable X; measures your earnings from the i'h toss:
1-P for Tail. You get a dollar if it comes
1 if the ith toss is Head
-1 otherwise
X;
Your profit (excess of heads over tails) after m tosses is
Sm = X1+
+ Xm:
Your progress in this game may be thought of as describing a random walk on the integers.
One way to think of this is on the line, or in one dimension: Let So = 0 signify the start at
the origin. Each toss is a step in which "Head" moves you one integer to the right from the
current position and "Tail" moves one integer to the left. Thus S; is the position of the walk
after the jth step (toss). The study of this interesting game reveals some surprising properties,
and it is another very significant example showing the power of generating functions.
• Positive Paths
two dimensions, i.e., in the plane: we use the x-axis for "time" and the y-axis for the
progress of the walk. The sequence of points {(0,0), (1, S1), (2, S2) . , (m, Sm)} describes
the start at (0, 0 [at time zero the game is even]; after step j our point has x-coordinate
j and y-coordinate S; = X1 + .… + X;. (S; is the the excess of heads over tails for the
first j tosses.) If we connect successive points by straight line segments we get what we
will call a random walk path with m steps (starting at the origin, ending at (m, Sm)).
There are 2m such paths (because each sequence of m tosses gives a distinct path). For
now we will take P = 1/2, so each of the 2m paths is equally likely.
The Ballot Theorem Another way to depict the above walk is in
•..)
We write
m = NH + NT
for the number of tosses, nH denoting the number of Heads and nT the number of Tails,
and observe that the score (our fortune) after m tosses is the difference between the
number of Heads and the number of Tails, or
— Пн — Пт:
Sm
Fix a positive integer k, 0 < k < m, and let us focus our attention on the paths that go
from (0, 0) to (m, k); i.e. Sm = k is the score after m steps.
Observation 1: Since Sm = NH
6.
NT
2nH – m, k is even if m is, and vice-versa.
Also note that the number of distinct paths which start at (0, 0) and end at (m, k) is
m
Nm,k
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