EE381-Lab 1 Random Numbers and Stochastic Experiments Solved

1.  Simulating  Coin Toss Experiments
 There are many ways to model stochastic experiments. The following two programs simulate the toss of a fair coin  N times, and calculate the experimental probability of getting heads (p_heads) or tails (p_heads). Both programs provide the same results, but they differ in the way the models are coded.

•        The first model is programmed in Python using "for loops". 

•        The second model makes use of the arrays, and it is computationally very efficient.

 

MODEL 1 
import numpy as np def coin(N):   

    heads, tails = 0, 0     for k in range(0,N):         coin=randint(0,2)         if coin==1:

            heads=heads+1         else:

            tails=tails+1

    #

    p_heads=heads/N     p_tails=tails/N

    print('probability of heads = ', p_heads)     print('probability of tails = ', p_tails)

     
 

 

MODEL 2 – MORE EFFICIENT CODE
import numpy as np def coin2(N):  

    coin=randint(0,2,N)     heads=sum(coin)     tails=N-heads

    #

    p_heads=heads/N     p_tails=tails/N

    print('probability of heads = ', p_heads)  print('probability of tails = ', p_tails)

 
 

 

0.2.  Roll of Two Fair Dice; Probability Mass Function (PMF)
 

This experiment models the roll of a pair of dice for N times. The sum each roll is recorded, and stored in vector "s". The probability of each possible outcome is calculated and plotted in a  "Probability Mass Function"  (PMF) plot

To create the plots, the simulation has been run for N=100000 times. 

 

SUM  OF  THE ROLLS  OF  TWO FAIR DICE
   import numpy as np    import matplotlib

   import matplotlib.pyplot as plt

# def sum2dice(N):

    d1=randint(1,7,N)     d2=randint(1,7,N)     s=d1+d2

    b=range(1,15) ; sb=size(b)     h1, bin_edges = histogram(s,bins=b)     b1=bin_edges[0:sb-1]     close('all')

    #

    fig1=plt.figure(1)     plt.stem(b1,h1)

    plt.title('Stem plot - Sum of two dice')     plt.xlabel('Sum of two dice')     plt.ylabel('Number of occurrences')     fig1.savefig('1 EE381 Proj Stoch Exper-1.jpg')

    #

    fig2=plt.figure(2)     p1=h1/N     plt.stem(b1,p1)

    plt.title('Stem plot - Sum of two dice: Probability mass function')     plt.xlabel('Sum of two dice')

    plt.ylabel('Probability')

       fig2.savefig('1 EE381 Proj Stoch Exper-2.jpg')

 
 

 

 

 

 

 

0.3.  Generating an unfair three-sided die
 

This example models the roll of a 3-sided die, with non-uniform probabilities. The die has three sides [1,2,3] with probabilities: [p p p1, 2, 3] = [0.3, 0.6, 0.1]. 

The experiment simulates the roll of the die for N=10,000 times, and the outcome of the N rolls is plotted as a stem plot. The stem plot verifies that the three sides of the die follow the required probabilities.

 

The following code will simulate a single roll of the three-sided die. The variable “d” represents the number after the roll.

 

import numpy as np

# n=3

p=array([0.3, 0.6, 0.1]) cs=cumsum(p)  cp=append(0,cs) r=rand() for k in range(0,n):     if rcp[k] and r<=cp[k+1]:

           d=k+1

 
 

 

 

The following code will simulate the rolling of the three-sided die for N=10,000 times and will plot the outcome as a stem plot.

 

import numpy as np import matplotlib

import matplotlib.pyplot as plt # def ThreeSidedDie(p):

    N=10000     s=zeros((N,1))     n=3

    # p=array([0.3, 0.6, 0.1])     cs=cumsum(p)      cp=append(0,cs)     #     for j in range(0,N):

        r=rand()         for k in range(0,n):             if rcp[k] and r<=cp[k+1]:

                   d=k+1         s[j]=d           

#

 
 

# Plotting             b=range(1,n+2)     sb=size(b)

    h1, bin_edges=histogram(s, bins=b)     b1=bin_edges[0:sb-1]     close('all')     prob=h1/N

    plt.stem(b1,prob)

 # Graph labels

    plt.title('PMF for an unfair 3-sided die')     plt.xlabel('Number on the face of the die')     plt.ylabel('Probability')       plt.xticks(b1)  

  
 

 

 

 

 



1.    Function for a n-sided die
 

Write a function that simulates a single roll of a n-sided die. The inputs and outputs of the function are: Inputs: 

•        The probabilities for each side, given as a vector p p p= [ 1, 2,Lpn]

Outputs:

•        The number on  the face of the die after a single roll, i.e. one number from the set of  integers {1,2,Ln} 

Note: The sum p p1 + +2 Lpn must be equal to 1.0, otherwise the probability values are incorrect.

Save  the function as: nSidedDie(p)

 

Test the function using the probability vector p p p= [ 1, 2,Lpn] which has been  given to you. To create a random number with a single roll of the die you must use the following command:  r=nSidedDie(p)

To validate your function, roll the die for N=10,000 times and plot the outcome as a stem plot.  

 

SUBMIT a report that follows the guidelines as described in the syllabus. 

Ø The section on RESULTS must include The PMF in the form of a stem plot  Ø The code must be provided in the appendix

 

2.    Number of rolls needed to get a "7" with two dice
 

Consider the following experiment: 

o   You roll a pair of fair dice and calculate the sum of the faces. You are interested in the number of rolls it takes until you get a sum of "7".  The first time you get a "7" the experiment is considered a "success". You record the number of rolls and you stop the experiment. 

o   You repeat the experiment N=100,000 times. Each time you keep track of the number of rolls it takes to have "success".