1 Coding a Poisson random variable

Suppose R (Python) can only generate uniform random variables. How can you take advantage of this and generate Poisson random variables?

2 Inverse transform method

Use the inverse transform method to generate an exponential random variable

3 The value of Pi

Inscribe a circle in a square. Estimate the value of \(\pi\) by computing the ratio of the number of times a uniformly chosen point on the square ends up in the circle.

4 Acceptance/Rejection

Using an exponential random variable, generate a normal random variable that is conditioned to be be positive; from here, adjust this result to get a normal random variable.

5 Total variatonal distance

Let \(X\) and \(Y\) be Poisson random variables with means \(\lambda > \mu\). Show that \[d_{TV}(X, Y) \leq 2 (1-\exp(\mu-\lambda))\]

6 Reversibility

Let \(P\) be transition matrix on a state space \(S\), and \(\pi\) be a probability measure on \(S\). We say that \(\pi\) is reversible for \(P\) if \(\pi_i p_{ij} = \pi_j p_{ji}\).

7 Simple card shuffling

Suppose that I have \(n=52\) cards, arranged in some initial order. Consider the following procedure: I choose two cards with probability \(1/ {n \choose 2}\), and then change their position; repeat.

8 The arrow of time

Consider a Markov chain on \(15\) states \(1,2,3,4, \ldots, 15\). Suppose that the probability of going from state \(i\) to \(i+1\) is \(0.9\), for all \(1 \leq i \leq 15\) and the probability of going from state \(i\) to \(i-1\) is \(0.1\) for all \(2 \leq i \leq 15\); in the case that \(i =1\), we allow the possibility of going to state \(15\) with probability \(0.1\), and if \(i=15\), the probability of going to state \(1\) is \(0.9\). Do you think this Markov chain has a stationary distribution; if so, what is it? Is this Markov chain reversible? Explain.

9 Endnotes