Question

Modified Random Walks: In the class Randomwalk, x_step and Y_step are generated from the same set of conditions. The direction is chosen randomly from the list \([1,-1]\) and the distance from the list \([0,1,2,3,4]\). Modify the values in these lists to see what happens to the overall shape of your walks. Try a longer list of choices for the distance, such as 0 through 8 , or remove the \(-1\) from the \(x\) or \(y\) direction list.

Short answer

Increasing distances spreads the walk, removing -1 from x results in a rightward walk trend.

Step-by-step solution

Understanding the Original Random Walk

In the original Random Walk, both x_step and y_step are determined by randomly selecting from the direction list \([1, -1]\) and the distance list \([0,1,2,3,4]\). This means each step can move in either a positive or negative direction with varying distances.

Modifying the Distance Choices

To start modifying the random walk, we change the distance list to \([0, 1, 2, 3, 4, 5, 6, 7, 8]\). The longer list of distances allows for potentially larger steps in each direction, which may increase the spread of the random walk.

Modifying the Direction Choices for X

Next, modify the x-direction list to \([1]\), meaning the steps will only be positive in the x-direction. This will result in the walk always moving forward along the x-axis, though distances will vary.

Observing the Modified Random Walk with Adjustments

After making these changes, run the modified random walk simulation. With the increased distances, the path is more spread out, and since the x-direction cannot be negative, the walk trends to the right.

Analyzing Results

Observe the resulting path. The walk has a wider east-west spread due to larger potential steps and shows a consistent rightward trend due to the removal of the negative x-direction option.

Key concepts

Random Walk Simulation

A random walk simulation involves creating a path using a series of steps where each step's direction and distance are determined randomly. In Python programming, you can simulate this by generating random numbers to choose from a designated list of directions and distances.
For a two-dimensional random walk, like the one in the exercise, both the x and y coordinates are updated at each step based on random selections. Originally, steps might be determined using both a variable for direction, selected from \([1,-1] \), and a distance variable picked from a list like [0,1,2,3,4]. These selections result in a zig-zag pattern as the steps alternate randomly across a grid.
Random walk simulations are a great way to learn about probability as you see how small, randomized decisions create unique paths every time you run the simulation.

Algorithm Modification

Algorithm modification is the process of changing elements of a procedure or method to observe different outcomes. In the context of this exercise, modifying the random walk involves altering the lists from which direction and distance are selected.
Consider changing the distance list from [0,1,2,3,4] to a longer set like [0,1,2,3,4,5,6,7,8], which increases the possibility for larger steps. This change makes the random walk more spread out as larger steps push the path further.
You can also adjust the direction choices. For example, by limiting the x-direction to only positive steps ([1]), the walk trends to the right, removing the chance of a leftward move. This particular modification showcases how restricting options impacts the final shape of the random walk path.

• Longer distance lists enhance spread.
• Direction modifications bias the walk to specific directions.

Probability and Statistics

Probability and statistics provide the backbone for understanding random walks, as they are processes governed by random chance. Each step in a random walk is an event with its own probability, which dictates how likely certain outcomes are.
When you run a random walk simulation multiple times, you observe varying paths due to the probabilistic nature of each step. This randomness allows us to explore statistical concepts, like variance and expectation, by analyzing how dispersed the paths become or the average position reached over repeated trials.
By modifying a random walk algorithm (as in this exercise), you directly alter the probabilities and expected outcomes. For instance, increasing possible step distances not only broadens the potential walkpath but also influences metrics like variance. Limited direction options skew the probability distribution towards particular orientations, offering a practical view of how different probabilistic inputs create diverse statistical scenarios.