Introduction

Plagiarism and collusion

Please familarize yourself with the following excerpt on plagiarism and collusion from the student handbook

By ticking the submission declaration box in Moodle you are agreeing to the following declaration:


Declaration: I am aware of the UCL Statistical Science Department’s regulations on plagiarism for assessed coursework. I have read the guidelines in the student handbook and understand what constitutes plagiarism. I hereby affirm that the work I am submitting for this in-course assessment is entirely my own. Moreover, I confirm that I have not used ChatGPT or other AI type tools to complete this assignment.

Anonymous Marking

Please do not write your name anywhere on the submission. Please include only your student number as the proxy identifier.

Questions

1.) A random point on a disc [10 points]

Suppose someone writes the following R code to generate a random point \(Z=(X,Y)\) on \(\mathbb{R}^2\):

R = 2*runif(1)
theta = 2*pi*runif(1)
X = R*cos(theta)
Y = R*sin(theta)
  • Modify the code so that it can produce as many independent samples of \(Z\) as you demand. [3 points]
  • Modify the code so that it rejects samples that lie outside a square of side length \(1\) centered at the origin. Call this resulting random point \(W\). [2 points]
  • Demonstrate by simulations that \(W\) is not uniformly distributed on the square. [3 points]
  • Prove that \(W\) is not uniformly distributed on the square. [2 points]

You may complete the question in Python, even though the above code was given in R

2.) Markov chains (I) [10 points]

The sample output of a Markov chain is available here.

  • Based reasonably on this output, generate and continue the Markov chain for another \(50\) steps. [5 points]

  • Explain why your procedure is reasonable and display the output of your generated \(50\) steps. [5 points]

3.) Markov chains (II) [10 points]

Let \(X\) and \(Y\) be independent Markov chains on a finite state space \(S = \{1,2,3\}\), with different transition matrices \(P\) and \(Q\), respectively. Suppose that the entries of \(P\) and \(Q\) are all positive. Also assume that \(X_0=1\) and \(Y_0=2\). Let \(T=\inf\{n \geq 1: X_n = Y_n\}.\)

  • Prove that \(\mathbb{E} T < \infty\). [4 points]
  • Let \(Y'_k = Y_k\) for all \(0 \leq k \leq T\) and set \(Y'_k = X_k\) for all \(k > T\). Is \((X,Y')\) a coupling of \(X\) and \(Y\)? Explain. [3 points]
  • Let \(c_n = d_{TV}(X_n, Y_n)\).
    • Does \(\lim_{n \to \infty}c_n\) exists? Explain. [2 points]
    • Can the limit be zero? Explain. [1 point]

4.) Poisson point processes [10 points]

Suppose someone wants simulate a Poisson point process of intensity \(2\) on the disc \(D\) of radius \(1\). However, they always delete/ignore and re-sample when they see that the computer returns zero points. Let \(\Pi\) represent the point process given by a kept sample.

  • What is the expected number of points of \(\Pi\)? [5 points]
  • What is the probability that \(\Pi\) has no points inside the inscribed square \(S \subset D\) of side length \(\sqrt{2}\)? Provide a numerical answer to four decimal places. [5 points]

Endnotes