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Simulation

Simulation#

Your mind makes it real.

—Morpheus, The Matrix

Simulation is a way of modelling random real-world processes. The idea relies on interpretting the results of random number generation as outcomes in an experiment. We have encountered this idea a few times in class already, but now that we have the concepts of probability in our catalogue of knowledge, we are ready to study simulation in detail.

Python has all the tools necessary to simulate experiments that would otherwise be tedious to perform. In this lab, we will explore a few techniques for conducting simulations to model the outcomes of random processes.

Instructions#

  1. Create a Python .py script named LASTNAME_FIRSTNAME_project_six.py in your Linux Files folder on your file system. You can do this by opening an IDLE session, creating a new file and then saving it. Replace LASTNAME and FIRSTNAME with your last and first name, respectively.

  2. Create a docstring at the very top of the script file. Keep all written answers in this area of the script.

  3. Read the Background section.

  4. Perform all exercises and answer all questions in the Project section. Label your script with comments where appropriate.

  5. When you are done, zip your script in a zip file named LASTNAME_FIRSTNAME_project_six.zip

  6. Upload the zip file to the Google Classroom Project Six Assignment.

Background#

Loops#

Loops are essential to the concept of simulation. A loop is a programmatic construct for repeating a block of instructions. As with many things in computer science and mathematics, this is easier to see than to say. Consider the following example,

for pet in ["cat", "dog", "hippopotamus"]:
        print(pet)

Output:

cat

dog

hippopotamus

Note

The pet variable is a local variable. It only exists within the for loop block. If you try to reference pet outside of the for loop, you will get an Undefined error.

This snippet of code uses a for-loop to iterate over the list containing the elements, dog, cat and hippopotamus. Each time it iterates over the list, it stores the current element in the pet variable. The body of the loop then uses the pet variable and prints it to screen. Each element of the list is output to screen; this is because the loop passes each element of the list to the pet variable and then executes the code in the body of the loop with that value of pet. Once there are no more elements in the list, the loop halts.

for loops can be used in conjunction with the range() function to iterate over sequences of integers starting at 0,

for i in range(5):
        print(i)

Output:

0

1

2

3

4

Note the output in the preceding example starts at 0 and ends at 4. The range() function iterates over all the natural numbers from 0 up to, but not including, the value inputted into the function.

Loops can be chained together to form nested loops,

for pet in ["dog", "cat"]:
        for owner in ["rory", "sophia", "sejal", "rachael", "lydia"]:

                print(owner, " has a ", pet)

Output:

rory has a dog

sophia has a dog

sejal has a dog

rachael has a dog

lydia has a dog

rory has a cat

sophia has a cat

sejal has a cat

rachael has a cat

lydia has a cat

The order of operations in a nested loop is important! For each iteration of the outer loop (the one iterating over pets), the inner loop (the one iterating over owners) is executed in its entirety.

Law of Large Numbers#

Definition#

Consider the experiment of flipping a fair coin ten times. We intuitively understand that, if the coin is truly fair, we should get an approximately even amount of heads and tails. However, the chances of getting exactly five heads and exactly five tails in one sequence of ten coin flips is small. Since each flip of the coin is independent of the previous coin flip, the fact we get a head on the first flip in no way influences the probability of getting a head on the second flip. The second coin flip has no responsibility to come out as tails if the first flip was heads.

The Law of Large Numbers gives a tool for understanding this result. First, we take the experiment of flipping a fair coin ten times and perform this experiment itself a large number of times, let us say 100 times. For each experiment of ten coin flips, we count the number of heads and the number of tails. If we then take the result of each repetition of the experiment and average them all together, the average value will be close to the true population value of five heads. The more times we repeat the experiment, i.e. the more samples we create to pool into the average value, the closer and closer the overall average becomes. In the limit, as the number of repetitions becomes infinite, the average value of the experiment equals its true value. In other words, if we replicate our experiment enough times, the average result will approximate its true population value. In this case, we should observe, after a large number of repetitions, the approximate proportion of heads to be 50% and the approximate proportion of tails to be 50%.

Python#

Using the Control Structures introduced in the preceding section and a few Built-In Functions we have used in previous projects, we can simulate the outcomes of random experiments and see the Law of Large Numbers in action.

The key idea is using random numbers to represent observations drawn from a given population. If we think of a random number between 0 and 1 as an observation, i.e. an outcome in an experiment, then we can use the random numbers to model the distribution of the experiment.

For example, let the event of getting a random number less than 0.5 correspond to the event of getting heads in a coin flip. Let the event of getting number between 0.5 and 1 correspond to the event of getting a tail. Then, by simulating random numbers between 0 and 1 and interpretting the results as outcomes of flipping a coin, we can derive a probability distribution for the experiment of flipping a coin any number of times.

The following code snippet simulates flipping a fair coin 10 times and stores the simulation in a list. It then takes the simulated distribution and displays a histogram to the user,

import random
import matplotlib.pyplot as mpl

# simulation parameters
no_simulations = 100
no_coins = 10
coin_prob = 0.5
# simulation results
head_dist = []

# simulation loop
for i in range(no_simulations):
        # resetting simulation variables
        sim_heads = 0

        # start simulation
        for j in range(no_coins):

                # simulating a single coin flip
                flip = random.random()

                # checking if simulated outcome = heads
                if flip >= coin_prob:
                        sim_heads += 1

        # end simulation

        print("simulation #", i)
        print("\t number of heads: ", sim_heads)

        head_dist.append(sim_heads)

# create histogram of heads distribution
## manually create bins to prevent weird histogram classes
bins = [ i + 1 for i in range(no_coins) ]
(fig, axes) = mpl.subplots()
axes.hist(head_dist, bins, ec="red",color="lightblue")
axes.set_xlabel("Number of Heads")
axes.set_ylabel("Frequency")
mpl.show()

Project#

  1. Consider the experiment of flipping 10 fair coins. Using the techniques described in the Background section, simulate flipping 10 coins. Perform the simulation 200 times.

    1. Calculate the mean and standard deviation of the simulated sampling distribution. In the docstring, write a sentence or two interpretting the meaning of these sample statistics in the context of the sampling distributions.

    2. Plot the results using a histogram. Label the axes appropriately. Ensure the histogram class limits are set to 0, 1, 2, …, 9, 10.

    Hint

    Use the bins argument on the hist() function to change the class limits! Refer to Histograms and Boxplots for more information on changing the histogram class limits!

    1. In the docstring, describe the simulated distribution in a few sentences. What value is the distribution centered around? What type of shape does the distribution have? Of what theorem in statistics is this an example?

    2. In the docstring, answer the following question: What would happen to the distribution if you increased the number of coins being flipped? What features mentioned in part c would change? What features would stay the same?

    Hint

    Test it out yourself by changing the number of coins in your code!

    1. In the docstring, answer the following question: What would happen to the shape of the distribution if you increased the number of simulations being performed? What features mentioned in part c would change? What features would stay the same?

    Hint

    Test it out yourself by changing the number of simulations in your code!

    1. In the docstring, answer the following question: What would happen to the shape of the distribution if you flipped an unfair coin, i.e. what would happen if you changed the probability of getting a head? What features mentioned in part c would change? What features would stay the same?

    Hint

    Test it out yourself by changing the probability of getting heads in your code!

    1. In the docstring, answer the following question: Based on the results of your simulation, what is the probability of observing 9 or more heads in a series of 10 coin flips?

  2. Approximately 2% of the world’s population has blonde hair. Consider the experiment of selecting 30 people at random from the world’s population and recording the number of people in the sample with blonde hair. Using the techniques described in the Background section, simulate the hair color of a sample of 30 people. Perform the simulation 500 times.

    1. Calculate the mean and standard deviation of the simulated sampling distribution. In the docstring, write a sentence or two interpretting the meaning of these sample statistics in the context of the sampling distributions.

    2. Plot the results using a histogram. Label the axes appropriately. Ensure the histogram class limits are set to 0, 1, 2, 3, …, 29, 30.

    3. In the docstring, describe the simulated distribution in a few sentences. What value is the distribution centered around? What type of shape does the distribution have?

    4. In the docstring, answer the following question: What happens to the variation in the simulation distribution as you increase the number of people sampled?

    5. In the docstring, answer the following question: What happens to the variation in the simulation distribution as you increase the number of simulations?

  3. Consider the experiment of rolling 10 six-sided die. Using the techniques described in the Background section, simulate 10 rolls of a six-sided die. Perform the simulation 500 times.

Hint

This one is easier to simulate if you use randint() instead of random()!

  1. Calculate the mean and standard deviation of the simulated sampling distribution. In the docstring, write a sentence or two interpretting the meaning of these sample statistics in the context of the sampling distributions.

  2. Plot the results using a histogram. Label the axes appropriately. Ensure the histogram class limits are set to 1, 2, 3, 4, 5, 6.

  3. In the docstring, describe the simulated distribution in a few sentences. What value is the distribution centered around? What type of shape does the distribution have? Of what theorem in statistics is this an example?

  4. In the docstring, answer the following question: what would happen to the shape of the distribution if you simulated rolling a 12-sided die instead of a six-sided die? What features mentioned in part c would change? What features would stay the same?

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