Pa starts from barn and every second takes one step. One day Pa starts to wander off, and notices that the grass has been mysteriously cut to resemble graph paper. Notice that after one step Pa is always exactly one unit away from the start. Let$$s assume that Pa wanders eastward from the initial location on the first step. How far away might Pa be from the initial location after the second step? John sees that with a probability of $0\mathrm{.}25$ Pa will be $0$ units away, with a probability of $0\mathrm{.}25$ Pa will be $2$ units away, and with probability of $0\mathrm{.}5$ Pa will be $2$ units away. On average Pa will be further away after two steps than after one step. If the second step is to the north or south, the third step will bring the farmer closer to origin half the time, further half the time. The third step will be away from the origin. If the second step is to the east the third step will be closer to origin a quarter of time, and further away three quarters of time.

It seems like the more steps Pa takes, the greater the expected distance from the origin. We could continue this exhaustive enumeration of possibilities and perhaps develop a pretty good intuition about how this distance grows with respect to the number of steps. Pa$$s wife Mi$-$Ma$,$ another grandparent of John$$s$,$ also likes to wander away randomly, but riding a mule. The mule goes South twice as often as any other direction. Lastly, John$$s hog Reg has an odd habit of wandering off too, but only randomly going east or west at each step, never north or south. Your Python program ought to model these two as well. Info on intructions:

simulate$()$

Define a function called simulate$()$ that takes three parameters:

A list of $$walk lengths$$ to simulate,

The number of trials $($how many times to do walks of the specified lengths$),$ and

Which walker we are modeling: $\text{'}$Pa$,\text{'}$Mi$-$Ma$,$ 'Reg$,$ or 'all$.$

For your convenience, define a main$()$ function $($that runs with conditional execution, per usual$).$ main$()$ should read the following three arguments from the command line:

A list of comma$-$separated $$walk lengths$$ to simulate $($there should be no spaces in this list$),$

the number of trials, or times to try each walk length, and

which type of walk we are modeling: $\text{'}$Pa$,$Mi$-$Ma$$ or $$Reg$$ or 'all$.$

If invoked with python$3$ random$\_$walk.py $100,100050$ all, your main$()$ function should call simulate$([100,1000],50,$ 'all'$).$

Assume the wanderer always starts each walk at $(0,0)$ in an infinite grid. On individual tests, your output values should be close to the examples given below, though they won$$t be exactly the same. The format of the output should be the same.

Pa random walk of $100$ steps

Mean $=8\mathrm{.}5$ CV $=0\mathrm{.}6$

Max $=19\mathrm{.}8$ Min $=1\mathrm{.}4$

Pa random walk of $1000$ steps

Mean $=31\mathrm{.}4$ CV $=0\mathrm{.}5$

Max $=57\mathrm{.}0$ Min $=1\mathrm{.}4$

Mi$-$Ma random walk of $100$ steps

Mean $=26\mathrm{.}7$ CV $=0\mathrm{.}4$

Max $=52\mathrm{.}6$ Min $=7\mathrm{.}6$

Mi$-$Ma random walk of $1000$ steps

Mean $=243\mathrm{.}4$ CV $=0\mathrm{.}2$

Max $=318\mathrm{.}2$ Min $=156\mathrm{.}2$

Reg random walk of $100$ steps

Mean $=7\mathrm{.}6$ CV $=0\mathrm{.}8$

Max $=22\mathrm{.}0$ Min $=0\mathrm{.}0$

Reg random walk of $1000$ steps

Mean $=33\mathrm{.}2$ CV $=0\mathrm{.}7$

Max $=86\mathrm{.}0$ Min $=0\mathrm{.}0$

In the output above:

Mean is the average distance over all walks of that length as the crow flies,

Max is the longest distance as the crow flies,

Min is the shortest distance as the crow flies, and

CV is the Coefficient of Variance, defined as the standard deviation divided by the mean.

Round all values to one decimal point, as shown.

All distances should be in steps.

plot$()$

Define a function called plot$()$ that plots a sample of final locations in a turtle window in order to visualize their behaviors. plot$()$ should expect no arguments. Your output should be similar to the picture below. Do only one plot: plot all three characters for $50$ trials on walk length $100.$ Don$$t plot the $1000$ walk length; it takes too long.

$.$guides$/$img$/$random$\_$walk

Details of plot$()$

In order to ensure that your output matches expected output, do the following:

Use turtle.shape$()($docs$)$ to change the turtle$$s shape $($the three shapes are 'circle$,$ 'square$,$ and $$triangle$).$

Use turtle.color$()($docs$)$ to change the turtle$$s color $(\text{'}$black$,$ 'green$,$ and $$red$).$

Use turtle.stamp$()($docs$)$ to plot individual points.

Use turtle.shapesize$()($docs$)$ to scale the shapes to half of their original size in both dimensions.

Use turtle.speed$()($docs$)$ and set it to the fastest speed if you have trouble with timing out.

Use a scale of five pixels $($for the turtle$)$ per step $($for Pa$,$ Mi$-$Ma$,$ or Reg$).$

Set the size of the screen to $300$x$400.$

When you are finished, call the provided save$\_$to$\_$image$()$ function

I have some code done. But I$\text{'}$m not sure what I need to change about it$.$ The rubric, expected$/$needed output, and my code so far is in the picture.