## Stochastics: Code for Martingales

Code for branching processes: branching_proc.R

Code for the Polya urn scheme: polya.R

Apr 23

Code for branching processes: branching_proc.R

Code for the Polya urn scheme: polya.R

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 at time . We assume that the Queen of Hearts starts on top, i.e. , and wish to compute the expected time for it to return to the top.

The probability transition matrix when there are a total of FIVE cards is:

*What is the expected time until the shuffler sees the Queen of Hearts on top of the deck when there are 52 cards?*

You may use the book, but you may not communicate with other students on this problem. Work smart! The solution does not require solving a 52-dimensional system of equations. (Hint: It may help to solve the 5-card case first and then attack the larger problem.)

This is your reminder that the course midterm is coming up soon. It will be in-class and closed book (and closed notes). You will not be allowed to use a calculator.

The midterm material will include everything we have covered in class from Durrett, Chapter 1.

Definitions, Statements of Theorems, Examples, Counterexamples: 50%.

Problem Solving: 50%

*Bayesian Inference and MCMC methods*, Math 7770

MWF 1:00 – 1:50 pm, Gibson 308

Office Hours, Gibson 427

Thu 10:00 am – 12:00 pm; Fri 11:00 am – noon.

Large Deviations Learning Group

Mon 3:00 – 3:50 pm

The Tulane Probability and Statistics Seminar

Wed 3:00 – 4:00 pm, Gibson Hall 414

6823 St Charles Ave

New Orleans, LA 70118

Phone: (504) 862-3426

E-mail: scott.mckinley@tulane.edu

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