PYTHON HOMEWORK 8 PSTAT 160A
PYTHON HOMEWORK 8
PSTAT 160A { S19
Professor Hohn
Instructions: Please note that you must work by yourself ! You will submit two les on Gau-
choSpace: (1) clear and concise explanations, graphics (if any), and results in PDF format (worth
40 points in total) and (2) your Python code in .py format (10 points). Note that if the grader
nds identical copies or very similar les, the grader cannot and will not grade them.
For Question #1, be sure to show all of your work! You may either (a) write at least a couple
sentences explaining your reasoning or (b) annotate your math work with brief explanations. Please
label any random variables or events that your use.
1. (10 points) (Without Python) After reading Fantastic Beasts and Where to Find Them, your
professor discovers that unicorns are equally likely to have 0, 1, 2, or 3 o spring. Unfortunately,
only one unicorn is left. That is, let (Zn)1 n=0 be a branching process that describes the popu-
lation of unicorns at time n. The o spring distribution is uniformly distributed on f0; 1; 2; 3g,
and Z0 = 1.
(a) Find the probability that the population of unicorns will go extinct by the second gener-
ation.
(b) Find the probability that the population of unicorns goes extinct.
2. (30 points) (With Python) In this Python exercise, you will compute the probability that a
branching process goes extinct. Be sure to annotate your code with short explanations of what
you are doing (worth 10 points).
Suppose (as above) that (Zn)1 n=0 is a branching process describing the number of unicorns in
a population with an o spring distribution that is uniformly distributed on f0; 1; 2; 3g:.
(a) Simulate 1 sample path of 10 steps of your Markov chain. Did the population go extinct?
Be sure to print your results. For example,
print('After 10 steps, the population of unicorns went extinct.')
(b) Simulate 1,000 sample paths of the Markov chain (Zn)1 n=0 as in part (a). Find the proba-
bility that the population goes extinct. Be sure to print your results.
print('The probability that the process goes extinct is %.4f.'
%prob_extinction)
(c) Take a screenshot showing your code and your results together (e.g. side by side).
Python Code Hints
numpy.linalg.matrix_power(P,3)
will give you P3 for a matrix P.
numpy.linalg.solve(A,b)
will solve the matrix equation Ax = b.
numpy.identity(n)
will give you the n n identity matrix.
numpy.matrix
will return a matrix from an array-like object or from a string of data. For example,
a = numpy.matrix([[1, 2], [3, 4]])
will give you the matrix
1 2
3 4
. It has certain special operators, such as * (matrix
multiplication) and ** (matrix power).
Change a matrix to a list via
nameOfMyMatrix.tolist()[0]
One way to make a bar graph is to use the library matplotlib.
matplotlib.pyplot.bar( x, height = y, align='center', alpha=0.5, color='g')
produces a bar graph with x-axis described by x and y axis described by y. Both x and y
are lists here.
matplotlib.pyplot.ylabel('Label Me')
creates a label for the y axis called Label Me.
matplotlib.pyplot.title('Title Me')
creates a title for the graph called Title Me.
matplotlib.pyplot.xticks(x, listOfNames)
will label the x axis tick marks with a list of names (e.g. Port numbers).
matplotlib.pyplot.show()
shows the graph/plot.
numpy.random.choice( alist, p = alist_prob )
will pick one element from the set alist using the probability distribution alist prob.
When Python indexes a list, the index of the list starts at 0. That is, to access the rst
entry of a list like a=[7, 8, 9], we need to write a[0].
alist = [1, 4, 7]
alist.append( 3 )
will add 3 to the end of your list. So, alist = [1, 4, 7, 3].
set()
creates a set { an unordered collections of unique elements.
aSet = set( ['Stark', 'Lannister', 'Greyjoy'] )
aSet.add( 'Baratheon' )
will add Baratheon to the set aSet.
alist = [1, 4, 7, 1, 4, 4]
alist.count( 4 )
returns 3, the number of times 4 occurs in the list alist.
Page 2
len (alist)
return the length (the number of items) of alist. alist can be a sequence (such as a list)
or a collection (like a set).
You may need to use a for loop or while statement in you code. See
http://www.openbookproject.net/books/bpp4awd/ch04.html for examples.
If you are using Python 2, you'll need to import division from Python 3 so that it acts
like Python 3 when using the / symbol in your calculations. If you are using Python 2,
write at the top of your py le,
from __future__ import division
You're importing the future!