Exercises, due Wed Sept 17. Jacod and Protter: 4.1, 5.20 and 7.11 4.1 (Poisson Approximation to the Binomial) Let be a Binomial probability with probability of success and the number of trials . Let . Show that Let and let change so that remains constant. Conclude that for small and large , , where .Continue reading “Probability Theory: Homework 2”

# Category Archives: Intro to Stochastic Processes

## Stochastics: Code for Martingales

Code for branching processes: branching_proc.R Code for the Polya urn scheme: polya.R

## Stochastics: HW 7, due Wed Apr 23

1. The Autocovariance problem posted last week.. 2. Consider the random variable and let be the event that or . Compute . Compute . Let be distributed like and let . Compute . Compute . 3. Let be a sequence of iid random variables and let . Let be the moment generating function of (andContinue reading “Stochastics: HW 7, due Wed Apr 23”

## 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 . An arriving particle that finds the counter free gets registered and then locks the counter for an amount of time . Particles that arrive while the counter is locked are not counted and have no effect.Continue reading “Stochastics: HW 6, due Wed April 16”

## 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 areContinue reading “Stochastics: HW 5, due Wed April 2”

## Reminder: No office hours today, Tuesday 3/18

## 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: Take home test problem, due Thursday at 5pm in LIT 460

We shuffle a deck by the following method: Take the top card off the top of the deck and place back into the deck in a position chosen uniformly at random. It is allowable that the top card will be put directly on top again. Let denote the position of the Queen of Hearts atContinue reading “Stochastics: Take home test problem, due Thursday at 5pm in LIT 460”

## 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).