Plagiarism and collusion

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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 \(t \to \infty\)?