MATLAB: An Introduction with Applications
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ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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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)?
* 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 \( \mathcal{P} \) for Head and probability \( q = 1 - \mathcal{P} \) for Tail. You get a dollar if it comes up Head and lose a dollar if it shows Tail. The coin will be tossed infinitely many times, and the random variable \( X_i \) measures your earnings from the \( i^{th} \) toss:
\[
X_i =
\begin{cases}
1 & \text{if the } i^{th} \text{ toss is Head} \\
-1 & \text{otherwise}
\end{cases}
\]
Your profit (excess of heads over tails) after \( m \) tosses is
\[
S_m = X_1 + \cdots + X_m.
\]
Your progress in this game may be thought of as describing a random walk on the integers.
- **Random Walk on the Line**: Imagine starting at \( S_0 = 0 \). Each toss moves you one integer to the right (Head) or to the left (Tail). Thus \( S_j \) is the position of the walk after the \( j^{th} \) step.
## Positive Paths - The Ballot Theorem
Another way to depict the above walk is in two dimensions, i.e., in the plane: using the x-axis for "time" and the y-axis for the progress of the walk. The sequence of points \((0, 0), (1, S_1), (2, S_2), \ldots , (m, S_m)\) describes the start at \((0, 0\) (at time zero the game is even). After step \( j \), the point has x-coordinate \( j \) and y-coordinate \( S_j = X_1 + \cdots + X_j \). Connect successive points by line segments to create a random walk path with \( m \) steps (start at \( (0,0) \), end at \( (m, S_m) \)).
There are \( 2^m \) such paths (each sequence of \( m \) tosses gives a distinct path). For \( \mathcal{P} = 1/2 \), each of the \( 2^m](https://content.bartleby.com/qna-images/question/6971cef0-9a3b-4ce4-80d1-ccd4d229c95d/9c1bd328-e2c6-4a88-b15d-80666a87693f/e7lz0hf_processed.png)
Transcribed Image Text:# A Random Walk on the Integers
We start with the coin toss game, which uses a coin with probability \( \mathcal{P} \) for Head and probability \( q = 1 - \mathcal{P} \) for Tail. You get a dollar if it comes up Head and lose a dollar if it shows Tail. The coin will be tossed infinitely many times, and the random variable \( X_i \) measures your earnings from the \( i^{th} \) toss:
\[
X_i =
\begin{cases}
1 & \text{if the } i^{th} \text{ toss is Head} \\
-1 & \text{otherwise}
\end{cases}
\]
Your profit (excess of heads over tails) after \( m \) tosses is
\[
S_m = X_1 + \cdots + X_m.
\]
Your progress in this game may be thought of as describing a random walk on the integers.
- **Random Walk on the Line**: Imagine starting at \( S_0 = 0 \). Each toss moves you one integer to the right (Head) or to the left (Tail). Thus \( S_j \) is the position of the walk after the \( j^{th} \) step.
## Positive Paths - The Ballot Theorem
Another way to depict the above walk is in two dimensions, i.e., in the plane: using the x-axis for "time" and the y-axis for the progress of the walk. The sequence of points \((0, 0), (1, S_1), (2, S_2), \ldots , (m, S_m)\) describes the start at \((0, 0\) (at time zero the game is even). After step \( j \), the point has x-coordinate \( j \) and y-coordinate \( S_j = X_1 + \cdots + X_j \). Connect successive points by line segments to create a random walk path with \( m \) steps (start at \( (0,0) \), end at \( (m, S_m) \)).
There are \( 2^m \) such paths (each sequence of \( m \) tosses gives a distinct path). For \( \mathcal{P} = 1/2 \), each of the \( 2^m
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