Exercises#

After completing the tutorial attempt the following exercises.

If you are not sure how to do something, have a look at the “How To” section.

Use the code written in the Modularisation Tutorial to obtain the average time until absorption from the first state of the Markov chains with the following transition matrices:
1. \(P = \begin{pmatrix}1/2 & 1/2 \\ 0 & 1 \end{pmatrix}\)
2. \(P = \begin{pmatrix}1/2 & 1/4 & 1/4\\ 1/3 & 1/3 & 1/3 \\0 & 0 & 1 \end{pmatrix}\)
3. \(P = \begin{pmatrix}1 & 0 \\ 1/2 & 1/2 \end{pmatrix}\)
4. \(P = \begin{pmatrix}1/2 & 1/4 & 1/4\\ 1/3 & 1/3 & 1/3 \\1/5 & 0 & 4/5 \end{pmatrix}\)

Modularise the following code by creating a function flip_coin that takes a probability_of_selecting_heads variable.

import random

def sample_experiment(bag):
    """
    This samples a token from a given bag and then
    selects a coin with a given probability.

    If the sampled token is red then the probability
    of selecting heads is 4/5 otherwise it is 2/5.

    This function returns both the selected token
    and the coin face.
    """
    selected_token = random.choice(bag)

    if selected_token == "Red":
        probability_of_selecting_heads = 4 / 5
    else:
        probability_of_selecting_heads = 2 / 5

    if random.random() < probability_of_selecting_heads:
        coin = "Heads"
    else:
        coin = "Tails"
    return selected_token, coin

Modularise the following code by writing 2 further functions:

get_potential_divisors: A function to return the integers between 2 and \(\sqrt{N}\) for a given \(N\).
is_divisor: A function to check if \(n | N\) (”\(n\) divides \(N\)”) for given \(n, N\).

import math

def is_prime(N):
     """
    Checks if a number N is prime by checking all that positive integers
    numbers less sqrt(N) than it that divide it.

    Note that if N is not a positive integer great than 1 then it does not
    check: it returns False.
    """
    if N <= 1 or type(N) is not int:
        return False
    for potential_divisor in range(2, int(math.sqrt(N)) + 1):
        if (N % potential_divisor) == 0:
            return False
    return True

Confirm your work by comparing to the results of using sympy.isprime.

Write a stats.py file with two functions to implement the calculations of mean and population variance.

Note that the mean is defined by:

\[ \bar x \frac{\sum_{i=1}^{N} x_i}{N} \]

The population variance is defined by:

\[ \sigma ^ 2 = \frac{\sum_{i=1}^{N} (x_i - \bar x) ^ 2}{N} \]

Use your functions to compute the mean and population variance of the following collections of numbers:
1. \(S_1=(1, 2, 3, 4, 4, 4, 4, 4)\)
2. \(S_1=(1)\)
3. \(S_1=(1, 1, 1, 2, 3, 4, 4, 4, 4, 4)\)
Compare your results to the results of using the statistics.mean, and statistics.pvariance. Note: the statistics library is part of the standard python library.

Python for mathematics

Exercises

Exercises#