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Exercise - Variance sample size dependence

Table of Contents

Introduction

In statistics it is always helpful to have a fairly large sample size and a great number of experiments to sharpen your statistical predictions. The following simple coin toss experiment will show how sample size and the number of experiments you do decrease the variance.

Requirements

Knowledge

To complete this exercise notebook you should possess knowledge about the following topics:

  • Variance

Python Modules

import matplotlib.pyplot as plt
import numpy as np
import scipy as sc
import scipy.stats as st
%matplotlib inline

np.random.seed(42)

Coin toss experiment

To simulate a coin toss we use the binomal distribution with probability 0.5 for a fair coin. The following expample shows 30 experiments with 10 throws each. You will see that often you get a number which is not equal to 5, which means that this specific experiment would suggest that the coin is not fair. So in order to test if the coin is fair or not you have to do some more throws and experiments and take the mean of all outcomes. 

print()
print('Example')
print('--------')
print('Do 30 experiments with 10 throws each and count the number of successes')
print()
CoinExp = np.random.binomial(10,0.5,30)
print(CoinExp)

Exercise

Complete the function cointossexp which simulates a coin toss experiment for a given setup. The function takes three parameters:

  • successprob - The probability of heads (0.5)
  • thrownumb - The number of throws in each experiments
  • expnumb - The number of experiments

Your functon should create three separate plots. On the x-axis, plot the number of throws. On the y-axis, plot: 1. the number of heads 2. the variance 3. the variance using plt.loglog scaling.

The values for number of heads and variance are to be taken as the mean across all experiments.

Comment on your results.

Hint: What meaning does the slope of your loglog plot have?

Your plots may look like the following: plot_example.png

def cointossexp(successprob,thrownumb,expnumb): 

    p = successprob               #set success probability (e.x. head)
    n = np.arange(1,thrownumb,1) #number n of throws
    N = expnumb                  #number N of experiments with n throws
    StrN = str(expnumb)          #used for Plot title
    
    
    CoinToss = []
    MeanHeads = []
    VarHeads = []
    
    #insert your code here
    
    return n,MeanHeads,VarHeads


MaxThrows = 100

exp1 = cointossexp(0.5,MaxThrows,2)
exp2 = cointossexp(0.5,MaxThrows,3)
exp3 = cointossexp(0.5,MaxThrows,8)
exp5 = cointossexp(0.5,MaxThrows,20)
exp6 = cointossexp(0.5,MaxThrows,30)
exp7 = cointossexp(0.5,MaxThrows,50)

Licenses

Notebook License (CC-BY-SA 4.0)

The following license applies to the complete notebook, including code cells. It does however not apply to any referenced external media (e.g., images).

exercise-variance-sample-size-dependence-new
by Oliver Fischer
is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Based on a work at https://gitlab.com/deep.TEACHING.

Code License (MIT)

The following license only applies to code cells of the notebook.

Copyright 2018 Oliver Fischer

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.