STAT 515
Homework #11 WITH SOLUTIONS
1. Suppose that Y1 , . . . , Yn is a simple random sample from N (, 1), where we assume that (log )/
has a t distribution on r degrees of freedom (this is the example we discussed briey in class on April
11). Assume th
STAT 515
Homework #11 WITH SOLUTIONS
1. Suppose that Y1 , . . . , Yn is a simple random sample from N (, 1), where we assume that (log )/
has a t distribution on r degrees of freedom (this is the example we discussed briey in class on April
11). Assume th
#hw 8 stat 515
#q1 part b
tpm=cbind(c(0,.5,.75),c(1/3,0,.25),c(.75,.25,0)
tpm
l=c(6,2,4)
cs1=1
cs2=1
jt1=NULL
jt2=NULL
for(i in 1:10000)
cfw_
jt1=c(jt1,rexp(1,l[cs1[i])
if(sum(jt1)>50) break
cs1=c(cs1,sample(x=c(1,2,3),size=1,prob=tpm[cs1[i],])
for(i in
STAT 515
Homework #8 WITH SOLUTIONS
1. The matrix exponential function is dened as
expcfw_M =
i=0
Mi
.
i!
However, this denition does not provide a suitable method for calculating expcfw_M for a given M . One simple
alternative is to use one of the two
STAT 515
Homework #7 WITH SOLUTIONS
1. Customers arrive at a post oce at a Poisson rate of 8 per hour. There is a single person serving customers, and
service times are exponentially distributed (and independent) with mean 5 minutes. Suppose that an arriv
STAT 515
Homework #6 WITH SOLUTIONS
1. Let N (t) be a non-homogeneous Poisson process with rate function (t).
(a) Fix some t > 0. Derive the conditional distribution of the arrival times S1 , . . . , SN (t) , conditional on
N (t) = n.
Solution: There are
STAT 515
Homework #5 WITH SOLUTIONS
A
This homework must be submitted electronically to ANGEL. I strongly encourage the use of L TEX.
1. Let X1 and X2 be independent exponential random variables with rates 1 and 2 , respectively. Let
X(1) = mincfw_X1 , X2
STAT 515
Homework #4 WITH SOLUTIONS
1. Suppose that in a branching process with X0 = 1, each individual produces some number of ospring that is
Poisson with mean 1, independently of all other individuals.
(a) What is the expected number of generations unt
STAT 515
Homework #3 WITH SOLUTIONS
1. Suppose that a population consists of a xed number, 2m, of genes in any generation. Each
gene is one of two possible genetic types. If any generation has exactly i (of its 2m) genes of
type 1, then for any 0 j 2m, th
STAT 515
Homework #1 WITH SOLUTIONS
1. A fair 6-sided die is rolled repeatedly. Let X equal the number of rolls required to obtain the
rst 5 and Y the number required to obtain the rst 6. Calculate
(a) E (X )
Solution: Since X is a geometric random variab
Stat 515: Stochastic Processes I
Final Examination
Spring 2012
Solutions
This nal exam is worth 30 points. You have 110 minutes. For full credit, you must explain all of your
work! Naturally, you may use any results that you know; you should not need to p
STAT 515
Homework #9 WITH SOLUTIONS
1. Customers arrive at a post oce at a Poisson rate of 8 per hour. There is a single person serving customers, and
service times are exponentially distributed (and independent) with mean 5 minutes. Suppose that an arriv
STAT 515
Homework #10 WITH SOLUTIONS
1. We wish to approximate = P (X > 4.5) where X N (0, 1). Suppose that q (x) is a normal density with
mean k , and suppose that X1 , . . . , Xn is a simple random sample from q ().
(a) Show that
=
1
n
n
2
2
i=1 I cfw_X
Homework 1, Stat 515, Spring 2017
Due Friday, January 20, 2017
1. Assume X Beta(, ). Find E(X | X < t) where t (0, 1).
2. The joint density of X and Y is given by
f (x, y) =
ex/y ey
,
y
0 < ,
0<y<
Show that E[X|Y = y] = y.
3. A gambler wins each game with
BRANCHING PROCESSES
1. GALTON-WATSON PROCESSES
Galton-Watson processes were introduced by Francis Galton in 1889 as a simple mathematical model for the propagation of family names. They were reinvented by Leo Szilard in the late
1930s as models for the pr
THE RENEWAL THEOREM
1. RENEWAL PROCESSES
1.1. Example: A Dice-Rolling Problem. Before the days of video games, kids used to pass their
time playing (among other things) board games like Monopoly, Clue, Parcheesi, and so on. In
all of these games, players
BROWNIAN MOTION
1. I NTRODUCTION
1.1. Wiener Process: Definition.
Definition 1. A standard (one-dimensional) Wiener process (also called Brownian motion) is
a stochastic process cfw_Wt t 0+ indexed by nonnegative real numbers t with the following
properti
POISSON PROCESSES
1. THE L AW OF SMALL NUMBERS
1.1. The Rutherford-Chadwick-Ellis Experiment. About 90 years ago Ernest Rutherford and
his collaborators at the Cavendish Laboratory in Cambridge conducted a series of pathbreaking
experiments on radioactive
CONTINUOUS-TIME MARKOV CHAINS
1. DEFINITION AND FIRST PROPERTIES
Definition 1. A continuous-time Markov chain on a finite or countable state space X is a family
of X valued random variables X t = X (t ) indexed by t 2 R+ such that:
(1)
(A) The sample path
HARMONIC FUNCTIONS AND MARKOV CHAINS
1. HARMONIC FUNCTIONS
Let (X n )n0 be a Markov chain with (finite or countable) state space X and one-step transition probabilities p (x , y ). A real-valued function h : X R is said to be harmonic at the site
x X if
X
Stat 515: Stochastic Processes I
Take-Home Final Exam
Spring 2012
April 21May 2, 2012
This exam is worth 10 points. You have 11 days. Electronic submission is required. Show all of your work
for full credit; answers submitted without supporting work will
Stat 515: Stochastic Processes I
Midterm
Spring 2012
WITH SOLUTIONS
This midterm is worth 20 points. You have 60 minutes. For full credit, you must explain all of your
work! Naturally, you may use any results that you know.
Problem 1. [8 points] A Markov
Stat 515: Stochastic Processes I
Midterm
Spring 2012
February 29, 2012
This midterm is worth 20 points. You have 60 minutes. For full credit, you must explain all of your
work! Naturally, you may use any results that you know.
Problem 1. [8 points] A Mark
STAT 515
Homework #11 WITH SOLUTIONS
1. Suppose that Y1 , . . . , Yn is a simple random sample from N (, 1), where we assume that (log )/
has a t distribution on r degrees of freedom (this is the example we discussed briey in class on April
11). Assume th
STAT 515
Homework #10 WITH SOLUTIONS
1. We wish to approximate = P (X > 4.5) where X N (0, 1). Suppose that q (x) is a normal density with
mean k , and suppose that X1 , . . . , Xn is a simple random sample from q ().
(a) Show that
=
1
n
n
2
2
i=1 I cfw_X
STAT 515
Homework #9 WITH SOLUTIONS
1. Customers arrive at a post oce at a Poisson rate of 8 per hour. There is a single person serving customers, and
service times are exponentially distributed (and independent) with mean 5 minutes. Suppose that an arriv