**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. ## Anonymous Marking Please do **not** write your name anywhere on the submission. Please include only your **student number** as the proxy identifier. # 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: `r format(Sys.time(), '%d %B %Y')` * [Rmd Source](https://tsoo-math.github.io/ucl3/2022-ica3-stat9-release.Rmd)