m/O {OL‘IA‘OM
Problem 1 (35 pts)
At the end of a month, a large retail store classiﬁes each of its customer’s accounts according to current
(0), 30—60 days overdue (1), 60—90 days overdue (2), more than 90 days (3). Their experience indicates
that the a
STAT/MATH 416
Spring 2016
Homework #1
Note: For this and future homework assignments, the chapters and respective questions
are from Introduction to Probability Models by Sheldon Ross, 11th edition. If you have
another edition, first check that the proble
STUDENT NAME . QUIZ #3
Attendance (5 pts). Remember to write your name above! To get full/partial credit Show work!
Questions are on BOTH sides of this page!
A plant species has red, pink, or white ﬂowers according to the genotypes RR, RW, and WW,
respect
Practice Problems for Midterm 2
1. Determine which of the following matrices are transition matrices and why:
2. Consider the Markov chain with states 0, 1, , 6 and transition probability matrix
0.7 0
0
0.1 0.2 0.3
0
0 0.5
0
0
0
0.6 0
0
0
0
0
0
0
0
For ea
STAT416 Homework 3
1. (Exercise #34, Chap. 2) Let the probability density of X be given by
cfw_
c(4x 2x2 ), 0 < x < 2
f (x) =
0,
otherwise.
(a) What is the value of c?
(b) Find P cfw_1/2 < X < 3/2.
2. (Exercise #43, Chap. 2) An urn contains n + m balls, o
Homework 4
1. (Exercise #74, a slightly simplified version) Let X1 , X2 , . . . , be a sequence of independent identically distributed random variables. Assume that there are no ties. We
say that a record occurs at time n if Xn > max(X1 , . . . , Xn1 ). T
Homework 6
1. #14 in Chap 4: Specify the classes of the following Markov chains, and determine
whether the states are transient or recurrent.
(a) The Markov chain has three states cfw_1, 2, 3
0 1/2
P =
1/2 0
1/2 1/2
and the transition prob. matrix is
1/2
Homework 2
1. (Exercise # 4 in Chap.2) Suppose a die is rolled twice. What are the possible values
that the following random variables can take on?
(a) The maximum value to appear in the two rolls.
(b) The minimum value to appear in the two rolls.
(c) The
Some Markov Chain Examples
Example 1. Coin 1 comes up heads with probability (0, 1) and coin 2 with probability
(0, 1). A coin is continually flipped until it comes up tails, at which time that coin is
put aside and we start flipping the other coin.
(a)
Counting Techniques
When all outcomes of an experiment are equally likely, the task of computing the
probabilities reduces to counting. In particular, if N is the total number of outcomes in
the sample space and N (A) is the number of outcomes contained i
Homework 5
1. (Exercise #3 in Chap.3) The joint probability mass function of X and Y , denoted
by p(x, y) is given by
p(1, 1) = 1/9,
p(2, 1) = 1/3,
p(3, 1) = 1/9
p(1, 2) = 1/9,
p(2, 2) = 0,
p(3, 2) = 1/18
p(1, 3) = 0,
p(2, 3) = 1/6,
p(3, 3) = 1/9.
Compute
Conditional Probability and Independence
For an experiment, information about the occurrence of one event B may cause us
to revise the probability of another event A. For example, let B be the event that a
randomly chosen individual is taller than 6 feet,
Homework 1
1. A box contains three marbles: one red, one green, and one blue. Consider an experiment that consists of taking one marble from the box then replacing it in the box
and drawing a second marble from the box. What is the sample space? If, at al
Solution
Stat 415: Homework 11
Question 5.21
E[time] = E[time waiting at 1] + 1/1 + E[time waiting at 2] + 1/2
And since
E[time waiting at 1] = 1/1
E[time waiting at 2] = (1/2 )
1
1 + 2
The last equation follows by conditioning on whether or not the custo
Denker
SPRING 2010
416 Stochastic Modeling - Assignment 2
SOLUTIONS
Problem 1:
E [X |Y = y ] means the expectation of the distribution of X when the outcome y of
the random variable Y is known.
First we calculate fX |Y (x|y ).
y
fY (y ) =
y
fX |Y (x|y ) =
Denker
SPRING 2010
416 Stochastic Modeling - Assignment 3
Solutions
Problem 1: (Problem 46, p.173) Show that
Cov (X, Y ) = Cov (X, E [Y |X ]).
We rst show that E [XE [Y |X ] = E [XY ]. Indeed if X and Y are discrete
E [XE [Y |X ] =
xE [Y |X = x]P (X = x)
Denker
Spring 2010
416 Stochastic Modeling - Assignment 4
Solution.
Problem 1: Let X0 be a rv with values in Z and let Zn (n 1) be independent, identically
distributed rv, also independent of X0 , taking values in cfw_1; 1. Prove or disprove whether
Xn ,
Denker
Spring 2010
416 Stochastic Modeling - Assignment 5
Solution
NOTE: All problems from assignemnts 15 may appear in the rst exam on
Feb 26!
Problem 1: (Example 4.6) A gambler beds at each play one third of his present fortune
rounded to the next full
Denker
Spring 2010
416 Stochastic Modeling - Assignment 6
Solution
Problem 1: (Problem 44, p. 271) Suppose that a population consists of a xed number
of genes, say m genes. If any generation has exactly i of its m genes of type 1, then the
next generation
Denker
Spring 2010
416 Stochastic Modeling - Assignment 7
Solution:
Problem 1: (Problem 55, p. 273) Consider a population of individuals each of whom
posesses two genes which can be either type A or type a. Call type A dominat and type a
recessive. Call a
Denker
Spring 2010
416 Stochastic Modeling - Assignment 9
SOLUTION
Problem 1: (Problem 12, p. 347) If Xi , i = 1, 2, 3, are independent exponential random
variables with rates i , respectively, nd
1. P (X1 < X2 < X3 ),
2. P (X1 < X2 | max(X1 , X2 , X3 ) =
Denker
Spring 2010
416 Stochastic Modeling - Assignment 10
SOLUTIONS:
Problem 1: (Problem 40, p. 352)
Let N1 and N2 be two independent Poisson processes. Show that N (t) = N1 (t) + N2 (t)
is a Poisson process and determine the rate of N .
Solution:
We sho
STAT/MATH 416
Spring 2016
Homework #2
To get full credit show all work!
Chapter 2: # 32, 34, 37, 50
Problem A:
For the random variable X from #2.34 find
a) the cumulative distribution function (cdf) (Note the cdf has to be specified for all real
values x,
STAT/MATH 416
Spring 2016
Homework #3
To get full credit show all work!
Chapter 2: # 54, 55(a,b), 55(c)[BONUS], 60, 65, 68
Problem A:
Let the joint pmf of X and Y be (, ) = 2 , = 1,2,3, = 1,2.
a) Find constant .
b) Find the marginal pmf of X and the margi
STAT/MATH 416
Spring 2016
Homework #6
To get full credit show all work!
Chapter 3: # 36, 37, 38, 44, 51, 56
Problem A:
A rat is trapped in a maze. Initially it has to choose one of two directions. If it goes to the right,
then it will wander around in the
STAT/MATH 416
Spring 2016
Homework #7
To get full credit show all work!
Chapter 4: # 1, 2, 3, 5, 6, 8, 10, 13[BONUS]
Problem A:
A Markov chain 0 , 1 , 2 , with state space cfw_0,1,2 has the following transition matrix
0
1 2
0.1 0.2
0
=
1 [0.9 0.1 0]
2
0.
STAT/MATH 416
Spring 2016
Homework #10
To get full credit show all work!
Chapter 5: # 1, 4, 5, 8, 12(a)[BONUS], 14, 15, 19
Problem A:
Let the pdf of random variable be
2
a)
b)
c)
d)
e)
() = cfw_
0
0
<0
Find the constant .
What is the distribution of ? Rem
Denker
SPRING 2010
416 Stochastic Modeling - Assignment 1
SOLUTIONS
Problem 1: (Problem 51, page 92) A coin, having probability p for landing heads, is
ipped until head appears for the r -th time. Let N denote the number of ips required.
Calculate E [N ].