91Ó°ÊÓ

In a random walk, you move according to rules with each move being determined by a random process. The simplest type of random walk is a one-dimensional walk where each move is either one step forward or one step backward on a number line. a. Start at 0 on the number line and flip a coin to determine your move. Heads means you take one step forward to the next integer, and tails means you take one step backward to the previous integer. What sequence of six tosses will land you on the number-line locations \(+1,+2,+1,+2,+3,+2\) ? (a) b. Explore a one-dimensional walk of 100 moves using a calculator routine that randomly generates \(+1\) or \(-1\). In list \(\mathrm{L} 1\), generate random numbers with 1 representing a step forward and \(-1\) representing a step backward. Describe what you need to do with list L1 to show your number-line location after every step. [r \([\) See Calculator Note 10A and 10B. 4] c. Describe the results of your simulation. Is this what you expected?

Step by step solution

01

Decode the movement pattern for part (a)

In part (a), we need to determine the sequence of coin tosses that land on the number line locations +1, +2, +1, +2, +3, +2. Start from position 0.

02

Step-by-step movements of part (a)

1. Start at 0. Toss 1 results in Heads, moving to 1. 2. Toss 2 results in Heads, moving to 2. 3. Toss 3 results in Tails, moving back to 1. 4. Toss 4 results in Heads, moving to 2. 5. Toss 5 results in Heads, moving to 3. 6. Toss 6 results in Tails, moving back to 2.

03

Compile toss sequence for part (a)

From the steps above, the sequence of tosses that lands on the specified positions is: Heads, Heads, Tails, Heads, Heads, Tails.

04

Generate random numbers for part (b)

In part (b), use a generator or a calculator to create a list of random steps where +1 denotes a step forward and -1 denotes a step backward. Store these in an array or list, such as L1.

05

Compute cumulative sum for step positions in part (b)

To find your position on the number line after each step in list L1, compute the cumulative sum of the entries. For example, if L1 is [1, -1, 1, 1, -1], the positions would be [1, 0, 1, 2, 1].

06

Analyze the results of simulation in part (c)

In part (c), reviewing the pattern of positions and whether it matches expectations involves checking how often you return to a central position and how much deviation occurs. Commonly, the walk will cluster around the start point (0) due to the nature of random walks.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

one-dimensional walk

Imagine you're walking along a straight path where every step you take is determined by chance. This is the essence of a one-dimensional random walk. In this simple setup, you can move either forward or backward, but not sideways or in any other dimension.

Unlike a regular stroll, where you decide where to go, in a one-dimensional walk, randomness dictates your path. You start at a specific position, such as zero on the number line, and each step is determined by flipping a coin or some other random mechanism. Heads could mean stepping forward, while tails might send you back.

This fundamental concept forms the basis of more complex theories in fields like physics, finance, and ecology, where understanding randomness and unpredictability is crucial.

number line

The number line is like an endless ruler with numbers spanning infinitely in both directions. It's a simple yet powerful way of visualizing numeric values, stretching from negative infinity to positive infinity, with zero sitting right at the center.

Your one-dimensional walk will take place entirely on this line. Imagine yourself at the origin, zero, ready to take steps forward or backward based on random decisions. Each toss of the coin represents a choice to either move towards the positive side or retrace towards the negative side.

This setup helps in visualizing the abstract idea of randomness. As you 'walk' on the number line, you're able to see how small decisions can lead to always changing outcomes, mimicking real-world uncertainties.

coin toss

Tossing a coin is probably one of the simplest and most well-known random processes. In the context of a one-dimensional walk, a coin toss offers a binary decision: moving forward or backward.

Here's how it works:

• Heads: Step forward, adding 1 to your current position.
• Tails: Step backward, subtracting 1 from your current position.

If you were to plot this out, each sequence of toss results would form a unique path along the number line.

This random process is a great way to simulate the unpredictability inherent in many natural and man-made systems.

simulations

Simulations allow us to model real-world processes or systems through experiments on a computer or other platform. When it comes to a random walk, simulations help expand on basic concepts by letting us see the outcome of numerous coin toss sequences without physically tossing a coin.

For instance, if you used a calculator to randomly generate sequences of numbers where '1' represents a forward step and '-1' indicates a step backward, you could simulate hundreds or thousands of steps in a few moments.

Through simulations, patterns in the random walk may emerge, such as the tendency to return towards the starting point. Such patterns are critical in helping us understand the behavior of complex systems that exhibit randomness in real life.