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Posts from the ‘Stochastics Homework’ Category

Stochastics: HW 7, due Wed Apr 23

1. The Autocovariance problem posted last week..

2. Consider the random variable

\displaystyle X := \left\{ \begin{matrix} 5 & \text{wp } 1/2 \\ 2 & \text{wp } 1/3 \\ 1 & \text{wp } 1/6 \end{matrix} \right.

and let A be the event that X = 2 or X = 5.

  • Compute \mathbb{E}[X \, | \, A].
  • Compute \mathbb{E}[X 1_{A}].
  • Let Y be distributed like X and let Z := X + Y.  Compute \mathbb{E}[Z \, | \, A].
  • Compute \mathbb{E}[Z \, | \, X].

3. Let \{X_1, \, X_2, \, \ldots\} be a sequence of iid random variables and let \mathcal{F}_n = \sigma(X_1, X_2, \ldots, X_n).  Let m(s) = \mathbb{E}[e^{sX_1}] be the moment generating function of X_1 (and hence of each X_i).  Fix s and assume m(s) < \infty.  Let S_0 = 0 and for n > 0,

S_n = X_1 + X_2 + \ldots + X_n.

Define M_n = m(s)^{-n} e^{s S_n}. Show that M_n is a martingale with respect to \mathcal{F}_n.

Stochastics: HW 6, due Wed April 16

Durrett 3.8 Counter processes. Suppose that arrivals at a counter come at times of a Poisson process with rate \lambda.  An arriving particle that finds the counter free gets registered and then locks the counter for an amount of time \tau. Particles that arrive while the counter is locked are not counted and have no effect. (a) Find the limiting probability that the counter is locked at time t.  (b) Compute the limiting fraction of particles that get registered.

Durrett 3.21 The city of Ithaca, NY, allows for two-hour parking in all downtown spaces.  Methodical parking officials patrol the downtown area, passing the same point every 2h.  When an official encounters a car, he marks it with chalk.  If the car is still there 2h later, a ticket is written.  Suppose that you park your car for a random amount of time that is distributed uniformly over (0,4) hours.  What is the probability you will get a ticket?

Durrett 4.2 A small computer store has room to display up to three computers for sale. Customers come at times of a Poisson arrival process at a rate of 2 per week. If a customer arrives and a computer is available, the customer will buy it. When the store has only one computer left they place an order for two more computers.  The order takes an exponentially distributed amount of time to be completed, with a rate of 1 per week. Of course, while the store is waiting for delivery, sales may reduce inventory from 1 to 0.  (a) Write a matrix for the transition rates Q_{ij} and solve \pi Q = 0 to find the stationary distribution. (b) At what rate does the store makes sales?

Durrett 4.9 A hemoglobin molecule can carry one oxygen or one carbon monoxide molecule.  Suppose that the two types of gases arrive at rates 1 and 2 and attach for an exponential amount of time with rates 3 and 4, respectively.  Formulate a Markov chain model with state space \{+ 0 -\} where + denotes an attached oxygen molecule, - denotes an attached carbon monoxide molecule, and 0 a free hemoglobin molecule.  Find the long-run fraction of time the hemoglobin molecule is in each of its three states.

Autocorrelation of a Stochastic Process (DUE WED APR 23)

Consider a sequence of random variables \{X_n\}_{n \geq 0} that take the values 5 and 1 with asymptotic frequencies \frac{2}{3} and \frac{1}{3} respectively.  We will consider some stochastic processes that produce such sequences and compute the one-step autocorrelation of these sequences.

The autocovariance function of a sequence is defined to be

\displaystyle \rho(i,j) = Cov(X_i, X_j)

where

\displaystyle Cov(X,Y) = E(XY) - E(X) E(Y)

  1. iid model. Suppose that the sequence is iid with P(X_n = 1) = 1/3 and P(X_n = 5) = 2/3.  Compute \rho(n, n + 1) for n \geq 0.
  2. Deterministic switching. Consider the sequence \{X_n\} = (1,5,5,1,5,5,1,5,5,1 \ldots).  Compute the “empirical” autocovariance function
    \displaystyle \hat \rho(1) = \lim_{N \to \infty} \frac{1}{N} \sum_{n=0}^{N-1} X_n X_{n+1} - \left(\frac{1}{N} \sum_{n=0}^{N-1} X_n\right) \left(\frac{1}{N} \sum_{n=1}^N X_n \right)
  3. Markov chain model. Suppose that \{X_n\}_{n \geq 0} is a Markov chain that takes the values 1 and 5 whose transition probabilities are given by the transition matrix
    \displaystyle P = \begin{pmatrix} 0.1 & 0.9 \\ p & (1 - p) \end{pmatrix}.

    • For what value of p will \{X_n\} satisfy the asymptotic frequencies given above?
    • Compute the one-step autocovariance function for this process assuming that the initial condition is drawn from the stationary distribution.

Stochastics: HW 5, due Wed April 2

This week’s problems are all directly from Durrett:

Chapter 2, #32, 33, 52, 58

Hint:  For the chicken crossing the road problem, you may find Theorem 2.10 useful.

UPDATE: It has been brought to my attention that the version of the textbook online has different numbers and one problem is missing altogether.  If you are working from the web version, the problem numbers are:

Chapter 2, #33, 34 (chicken crossing the road), 48 (preventative maintenance) and the following,

#58 For a Poisson process N(t) with arrival rate 2, compute: (a) \mathbb{P}(N(2) = 5), (b) \mathbb{P}(N(5) = 8 \, | \, N(2) = 3, and (c) \mathbb{P}(N(2) = 3 \, | \, N(5) = 8).

UPDATE:

HW 5 Solutions (pdf)

Stochastics: Some suggested theoretical exercises

These exercises are not required, but they are recommended for graduate students and others enrolled in the 5000-level of this class.

Theoretical Exercises (pdf)

Stochastics: Suggested Problems for Chapter 2

Practice with exponential random variables
2.1 – 2.3, 2.32

Comparison of multiple exponentially distributed random events
2.7, 2.8

Queueing Theory
2.9, 2.10

The Poisson Process
2.22, 2.23, 2.25, 2.27, 2.33

Stochastics: HW 4 Solutions

HW 4 (pdf)

HW4 Solutions (pdf)

In these solutions I reference the use of Wolfram Alpha, (special section on matrix functions here).

hw4 wolfram alpha exmaple

Stochastics: Challenge Problem

If you’re struggling with understanding the notion of detailed balance, try out this quick challenge problem.

  • Consider RW on the graph pictured below.  For what value(s) of p, does there exist a measure \mu that satisfies the Detailed Balance condition.

Detailed Balance Example

 

Stochastics: Solutions to HW 3

Note! There was a mistake in the earlier version of these solutions.  This version is current as of 5 pm Monday, Feb 24.

MAP 4102 Homework 3 (pdf)

M4102 HW 3 Solutions (pdf)

Stochastics: Suggested Problems for Exam 1

Getting things kicked off on the new website. Here is a list of interesting problems to try from Durrett’s book.  Please use the comments feature to ask questions and to start discussions about these problems. I’ll chime in whenever I can.

Stationary distributions and limit distributions
1.20, 1.25, 1.38, 1.41, 1.44

Random walks on graphs
1.49-1.51

Hitting times; hitting probabilities
1.59, 1.60

Infinite state spaces
1.70, 1.73