Plagiarism and collusion

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1.) Parameter estimation [10 points]

Let \(X_1, X_2, \ldots\) be iid exponential random variables with unknown rate \(\lambda>0\). Suppose we only get to keep track of whether \(X_i\) is in the unit interval or not; that is, we observe the random variables \(Y_i = \mathbf{1}[ 0 \leq X_i \leq 1]\). Find a reasonable estimator for \(\lambda\) and provide justification that your estimator is good.

2.) The area of a random triangle inside a disc [10 points]

Demonstrate by simulations that if three points are sampled independently and uniformly at random inside a disc of radius \(1\), then the expected area of the corresponding triangle is \(\tfrac{35}{48\pi}\).

3.) Markov chains [10 points]

  • Suppose \(X\) is irreducible and aperiodic finite state Markov chain on a state space \(S\), containing the states \(a,b\) and \(c\). Consider the associated sums given by \[T_n:= \frac{1}{n+1} \sum_{k=0} ^n \mathbf{1}[X_{k+2} =c, X_{k+1}=b , X_{k} =a].\] Guess the limit for \(T_n\) as \(n\to \infty\). Provide evidence (not necessarily a proof) for your guess. [5 points]

  • Demonstrate your guess, via simulations, using the Markov chain with transition matrix \[ \begin{equation*} P = \begin{pmatrix} 2/6 & 3/6 & 1/6 \\ 1/4 & 1/4 & 1/2 \\ 1/8 & 1/4 & 5/8 \end{pmatrix}. \end{equation*} \] [5 points]

4.) Point processes [10 points]

Consider the point process on the unit interval \([0,1]\) that is a sampled by placing \(N=10\) points independently and uniformly on the interval. Let \(X\) and \(Y\) be the number of points on the two disjoint halves of the interval.

  • Are \(X\) and \(Y\) independent? Explain. [3 points]
  • What is the \(\mathbb{E} X\)? [2 points]
  • What is the probability mass function for \(X\)? [3 points]
  • What is the distribution of location of the point closest to the right endpoint of the interval? [2 points]