Lab 11

In this lab, we explore the use of random numbers in programming using numpy.random . Random numbers are useful in modeling stochastic processes as simple as flipping a coin or rolling dice, or as advanced as quantum mechanical probabilities. Random numbers can be used to generate simulated data sets with uncertainties. They are also commonly utilized to perform integration in what is known as the Monte Carlo technique.

Exercise 1: Uniformly distributed random numbers

Numpy provides several random number generators. In this example we focus on rand which return a random number from 0-1.
#! /usr/local/bin/python3  

import numpy as np

x1 = np.random.rand()

x2 = np.random.rand(5)

a, b = 1, 20
x3 = (b-a)*np.random.rand(20) + a

explaining code

  • look at the content of x1,x2,x3. What do they return?
  • Use rand to return 50 random numbers between 10 and 30.

    Exercise 2: Random distribution of integers.

    In this example we focus on randint which returns a random integer number.

    #! /usr/local/bin/python3  
    import numpy as np
    x1 = np.random.randint(1,100,10)
    explaining code

  • Use randint to return the results of a series of "coin tosses". If randint returns 0 than the coin toss is head and if randint returns 1 than the coin toss is one.

    Exercise 3: probability distribution

    In addition to plotting the histogram " count vs. variable" you can also plot the probability histogram " probability of count vs. variable." This is also done by matplotlib function hist Keyword arguments weights as shown below.
    #! /usr/local/bin/python3  
    import numpy as np
    import matplotlib.pyplot as plt
    a, b = 0, 10
    x1 = (b-a)*np.random.rand(100) + a
    weights1 = np.ones_like(x1)
    weights2 = np.ones_like(x1)/float(len(x1))
    n1, b1, p1= plt.hist(x1,bins=10,range=(0.0,10.0),histtype='stepfilled',color='green', weights=weights1,alpha=0.5)
    plt.title('Count distribution')
    n2,b2,p2 = plt.hist(x1,bins=10,range=(0.0,10.0),histtype='stepfilled',color='blue',weights=weights2,alpha=0.5)
    plt.title('Probability distribution')
    explaining code
  • find the sum of the bin values for the probability distribution. What should it equal to. Note that for an array b the sum of its elements can be achieved by using sum(b).
  • Histogram the result of ex2.

    Mandatory Assignment, due to Monday noon.

    Problem 1

  • part-a: Write a function dice2 that generate two random numbers and use these to assign each die a number between one and six with equal probability. The input of the function dice2 should be N the number of times it will generate two random numbers. Of course your histogram should allow integer entries between (at least) 2 and 12. "Roll" 2 dice N time where N = 10,000 times keeping track of all the sums of each set of rolls in a list. Then sum these two numbers and histogram the results. save histogram in dice2.png
  • plot the probability distribution for part-a's histogram with title dice2prob.png
  • part-b: Write a function dice3 using 3 dice and plot the resulting histogram.
  • plot the probability distribution for the part-b's histogram with title dice3prob.png
  • submit your python script and your plots dice2.png dice3.png dice2prob.png dice3prob.png . Add a title to each plot. Use histogram type stepfilled. With a color other than black. Write in your code which sum of dice2 would you bet on.

    Problem 2

    For this problem, we will use the Monte Carlo integration technique to compute the value of π

    Start with the code , which generates two random numbers and generates a scatter plot of the results. Run the code as is and see the output.

    Your task is to determine ( nhit )The number of points inside the blue circle of radius r=0.5 on the scatter plot. From that determine π. Submit your modified code also contains a function binomialError which will take the value of nhit and returns two values:
  • The fraction f of the number of points hitting the circle to the total number of thrown points
  • The uncertainty in the value of f. Note that if x = const. * A than the error in x = const. * error in A.
    should print your answer to the terminal for the value of π and the expected uncertainty in π.

  • Note that the origin (0,0) is not at the center of the circle. This hint is important when selecting events inside or outside your object (the circle)
  • From lecture
    area \ of \ object = area \ of \ rectangle \times \frac{number \ of \ points \ inside \ the \ object}{total \ thrown \ number \ of \ points \ inside\ rectangle}

  • what is the value of the
    \frac{ area \ of \ object} { area \ of \ rectangle}
    . How does it relate to the value of π knowing the radius of the circle and the area of your square.

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