Plagiarism and collusion

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1.) [10 points] Apery’s constant

  • [2 points] Prove (using calculus) that

\[\zeta(3) = \sum_{n=1} ^{\infty} \frac{1}{n^3} < \infty.\]

  • [6 points] Consider the probability distribution \(\mu\), where \[\mu(n) = \frac{1}{\zeta(3)n^3}.\] Use the Metropolis algorithm to sample from \(\mu\), so that you do not need to know the value of \(\zeta(3)\).

  • [2 points] Now that you can sample from \(\mu\), use simulations to estimate the value of \(\zeta(3)\).

2.) [15 points] Bayesian statistics and MCMC

We say that a positive continuous random variable \(X\) has the inverse gamma distribution with parameters \(\alpha >0\) and \(\beta >0\) if it has pdf given by \[(y; \alpha, \beta) \mapsto \frac{\beta^{\alpha}}{\Gamma(\alpha)} y^{-\alpha -1} e^{\tfrac{-\beta}{y}} \mathbf{1}[y >0],\] where \(\Gamma\) is the usual Gamma function.

We say that a positive continuous random variable \(W\) has the Scaled-Weibull distribution with shape parameter \(k\) and scale parameter \(\theta >0\) if it has pdf given by \[(w_1; k,\theta) \mapsto \mathbf{1}[w_1 >0]\frac{k w_1^{k-1}}{\theta} \exp[ - \tfrac{w_1^{k}}{\theta } ] .\]

  • [2 points] Let \({W} = (W_1, \ldots, W_n)\) be a random sample from the Scaled-Weibull distribution with known shape parameter \(k\) and unknown scale parameter \(\theta >0\). Show that \(t({W}) := \sum_{i=1} ^n W_i ^k\) is a sufficient statistic for \(\theta\).

  • [3 points] Fix \(k >0\). Let \({X} = (X_1, \ldots, X_n)\) be a random sample where the conditional distribution of \(X_1\) given \(\Theta = \theta\) has the Scaled-Weibull distribution with shape parameter \(k\) and scale parameter \(\theta\), and \(\Theta\) has the inverse gamma distribution with parameters \(\alpha\) and \(\beta\). Given sample data \(x=(x_1, x_2, \ldots, x_n)\). Compute the posterior distribution \(s(\theta|t(x))\) up to constant factors.

  • [3 points] Identify the distribution of \(s(\theta|t(x))\).

  • [4 points] Now pretend you could not identify it, and could not deduce exact constant factors. For the simple case, where \(\alpha =2\), \(\beta=3\), \(n=3\), and \(x_1=2, x_2=4, x_3=6\), sample from \(s(\theta|t(x))\) using the Metropolis algorithm; also take \(k=1\).

  • [3 points] Plot independent samples in a probability histogram and compare with the true result.

3.) [15 points] A Poisson process process on a perimeter of a semi-circle

Let \(\Gamma\) be a homogeneous Poisson point process of intensity \(2\) on the upper half of the circle given by \(x^2+y^2 =1\). Here, \(\Gamma\) is not the Gamma function. Consider the point process \(\Upsilon\) given by the projection of \(\Gamma\) onto the \(x\)-axis; that is, if \(\Gamma\) had \(n\) points and they are given by \((x_1, y_1), \ldots, (x_n, y_n)\), then the points of \(\Upsilon\) are just the \(x\)-coordinates \(x_1, \ldots, x_n\).

  • [5 points] Write code to simulate \(\Gamma\) and \(\Upsilon\). Graphically display a sample realization of these point processes.

  • [5 points] Demonstrate using simulations that \(\Upsilon\) is not a homogeneous Poisson point process on \([-1,1]\).

  • [5 points] Show analytically that \(\Upsilon\) cannot be a homogeneous Poisson point process on \([-1,1]\).

4.) [15 points] The transition rate matrix

You are given the the sample data from an irreducible continuous-time Markov chain. The sample data includes the jump times \((0,j_1, \ldots, j_n)\) and states \((s_0, s_1, \ldots, s_n)\); here at time \(j_i\) the Markov chain jumps into state \(s_i\) and stays there until the next jump which occurs at time \(j_{i+1}\).

  • [8 points] When \(n\) is large, give a method for estimating the transition rate matrix, also referred to as the \(Q\) matrix. Explain why your estimate is reasonable.

  • [7 points] Import the data from the file Q.txt and use this data and your method above to estimate the \(Q\) matrix.

5.) [10 points] Queues

Suppose you have Poisson arrivals, with intensity \(6\). You are given the following two options. Option 1: we treat it like a \(M(6)/M(8)/1\) system- the items are served by exponentially at rate \(8\). Option 2: each item is painted red or blue independently with probability \(\tfrac{1}{2}\); the coloured items report to different queues, with the red items are served exponentially at rate \(4\), and the blue items served exponentially at rate \(4\).

  • [5 points] Run simulations to identify the stationary distributions of the items in each of the two options. Which option, on average, has more items in it?

  • [5 points] Which option is better, from the items/customers perspective? Explain, analytically.