# Introduction

• Made available UCL Week 16
• Due UCL Week 24; 9 February 2023, 13:00 London time.
• You are to work in groups of 2-4; there should be no overlap with your ICA2 groups.
• This ICA consists of questions, worth 55 points;
• another 5 points will be given based on presentation, so that the total available points is 60
• This ICA is worth 30 percent of your grade for this module.
• Many questions will require you to write and run R/Python code.
• The ICA must be completed in R Markdown/Juptyer Notebook and typeset using Markdown with Latex code, just like the way our module content is generated. You can choose to use either R or Python.
• Please hand in the html file and the Rmd/ipynb source file.
• As usual, part of the grading will depend on the clarity and presentation of your solutions.
• You are to do this assignment by yourselves, without any help from others.
• You are allowed to use any materials and code that was presented so far.
• Do not search the internet for answers to the ICA.
• Do not use any fancy code or packages imported from elsewhere.

## 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.

# Questions

## 1.) An ellipse and a circle [5 points]

By running simulations of uniform random variables, estimate the area in the intersection of an ellipse and circle, with equations given by: $\frac{x^2}{25} + y^2=1$ and $(x-4)^2 +(y-2)^2 =4.$

## 2.) Bivariate normals [5 points]

Let $$Z_1, Z_2, \ldots$$ be independent standard bivariate normal random taking values in $$\mathbb{R}^2$$. A lecturer claims that by simply knowing whether $$Z_i$$ is in the unit circle $$x^2 + y^2 =1$$, for a large number for $$i=1,2,3, \ldots$$, they can estimate the value of $$\exp(1) \approx 2.71828..$$. Is this claim true or false? Explain.

## 3.) Markov chains [10 points]

Let $X = (X_0,X_1 \ldots, X_N)$ be $$(N+1)$$-steps of an aperiodic irreducible finite state Markov chain with a transition matrix $$P$$ and state space $$S$$. Consider the sequence of random variables given by $Y := (Y_0, Y_1, \ldots, Y_N)= (X_N, X_{N-1} \ldots, X_0).$

• [3 points] Show that $$Y$$ is $$(N+1)$$-steps of a Markov chain that is not necessarily time-homogeneous.
• [3 points] Show that if $$X_0$$ has the stationary distribution for $$P$$, then $$Y$$ is a time-homogeneous Markov chain;
• [4 points] in this case, identify the $$\lim_{N \to \infty}\mathbb{P}(X_0 = j | X_N = i)$$ for states $$i,j \in S$$.

## 4.) Poisson processes [5 points]

Given $$\lambda_N > \lambda_Q >0$$, show that there exists Poisson processes $$N$$ and $$Q$$, with rates $$\lambda_N$$ and $$\lambda_Q$$, respectively, such that $$N(t) \geq Q(t)$$ for all $$t \geq 0$$, and such that the first arrival for $$N$$ always comes strictly before the first arrival for $$Q$$.

## 5.) Continuous-time Markov chains [10 points]

Consider a continuous-time Markov chain $$X$$ on three states $$\{1,2,3\}$$ with transition rate matrix $Q= \begin{pmatrix} -5 & 2 & 3\\ 1 & -4 & 3 \\ 1 & 2 & -3 \end{pmatrix}.$

Suppose that $$X$$ is started at $$1$$, so that $$X(0)=1$$. Let $$Y= \inf \{t \geq 0: X(t) \not = 1\}$$. Let $$T = \inf \{t> Y: X(t) =1 \}$$.

• [5 points] Compute (analytically, exactly), $$\mathbb{E}(T)$$.
• [5 points] Compute, by simulating the Markov chain, $$\mathbb{E}(T)$$.

## 6.) Infinite servers [20 points]

Consider the following system where items arrive as a Poisson process with intensity $$\lambda >0$$, and there are an infinite number of servers, each of which gives service time corresponding to a continuous random variable with cdf $$F$$, independently, to the customers. Assume that a random variable with cdf $$F$$ has finite mean.

• [10 points]
• [6 points] Code this queue in the special case that $$\lambda =1$$ and $$F$$ corresponds to the uniform distribution on interval $$[1,2]$$.
• [2 points] In particular, for $$t=100$$, plot a probability histogram of $$Q(t)$$, the number of items in queue at time $$t$$.
• [1 point] What is the mean and variance?
• [1 point] Identify the distribution.
• [3 points] Apply Little’s law to analytically obtain the (long term) average number of items in the queue.
• [7 points] In the following question, we will analytically compute the distribution for $$Q(t)$$; do not assume that $$\lambda=1$$ or that $$F$$ corresponds to the uniform distribution. Let $$N(t)$$ be the number of arrivals by time $$t\geq 0$$.
• [2 points] Suppose there is only one arrival by time $$t>0$$; that is, one item has arrived in the time interval $$[0,t]$$. Show that $p:=\mathbb{P}(Q(t) =1 | N(t)=1 ) = \frac{1}{t}\int_0^t[1- F(z)]dz.$ Hint: recall that if there is only one Poisson point in $$[0,t]$$, then it is uniformly distributed in that interval.
• [2 points] Compute and identify the pmf given by $$\mathbb{P}(Q(t) =k | N(t)=n)$$ for $$0 \leq k \leq n$$ and fixed $$n \geq 1$$.
• [2 points] Compute and identify the pmf for $$Q(t)$$.
Hint: if you use all your knowledge about Poisson processes, you may be able to avoid a minor calculation.
• [1 point] What is the limiting pmf for $$Q(t)$$ as $$t \to \infty$$?

# Endnotes

• Q2: the word standard is used to mean that the two components are independent are mean-zero with unit-variance.

• Q4: the word strictly is added to clarify.

• Version: 17 December 2022

• Rmd Source