General rules

Additional rules

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Questions

1 (5 points)

For each \(\epsilon \in [0,\tfrac{1}{4}]\), find a coupling of Bernoulli variables \(X\) and \(Y\) with parameter \(\tfrac{1}{2}\) such that \(\mathbb{P}(X \not = Y) = \epsilon\).

2 (10 points)

Let \(f_1(x) = 3x^2\) and \(f_2(x) = 4x^4\). Define the random variable \(X\) in the following way. Choose a point \(Z = (U, V)\) uniformly at random in \([0,1] \times [0,4]\). If \(Z\) lies under both the curves \(f_1\) and \(f_2\), then set \(X=U\); otherwise we independently repeat this procedure until \(X\) is defined.

  • (5 points) By running simulations plot of a histogram representing the pdf for \(X\).
  • (5 points) Compute the exact formula for the pdf for \(X\). Overlay your pdf for \(X\) with the histogram.

3 (5 points)

Let \(U_1\) and \(U_2\) be independent uniform random variables distributed on \([0,1]\). Consider the points \(T_1 = U_1\) and \(T_2 = U_1 + U_2\). Suppose the interval \([0,1]\) contains exactly one point.

  • (3 points) Compute, by hand, the distribution of this point.
  • (2 points) Check your answer by doing simulations.

4 (10 points)

Consider the Markov chain \(X\) on five states \(\{1,2,3,4,5\}\) started at \(1\) with the following transition matrix:

P <- matrix(c(1/8, 1/4, 1/2, 1/8,0, 
              1/4, 1/4,0,0,1/2, 
              0.1,0.2,0.3, 0, 0.4, 
              0,0,0,1/2, 1/2, 
              1/3, 1/3, 0, 0, 1/3), nrow =5)
P <-t(P)
P
##           [,1]      [,2] [,3]  [,4]      [,5]
## [1,] 0.1250000 0.2500000  0.5 0.125 0.0000000
## [2,] 0.2500000 0.2500000  0.0 0.000 0.5000000
## [3,] 0.1000000 0.2000000  0.3 0.000 0.4000000
## [4,] 0.0000000 0.0000000  0.0 0.500 0.5000000
## [5,] 0.3333333 0.3333333  0.0 0.000 0.3333333

Consider \(S_n = \sum_{k=0}^n\mathbf{1}[X_{k}=1,X_{k+3}=5]\).

  • (5 points) Make an educated guess, by extending known results, about the convergence of \(S_n/(n+1)\), as \(n \to \infty\).
  • (5 points) Demonstrate your guess is correct via simulation.

5 (10 points)

Consider the process \(X = (X_0,X_1,\ldots)\) defined in the following way. Let \(U_0, U_1, \ldots\) be iid random variables that are uniformly distributed in \([0,1]\). We set \(X_0=1\) and \(X_1=1\). We also have a function \(\phi:\{1,2\}^2 \times [0,1] \to \{1,2\}\) such that for \(n\geq 0\), we have \(X_{n+2} = \phi(X_{n+1}, X_{n}, U_{n+2})\). Furthermore, we are given the probabilities \(\mathbb{P}(X_{n+2}=k | X_{n+1} =j,X_n =i ) := p_{ijk}\), in the following table.

ij k=1 k=2
i=1,j=1 0.3 0.7
i=1,j=2 0.2 0.8
i=2,j=1 0.1 0.9
i=2,j=2 0.4 0.6

Consider \(a_n:=\mathbb{P}(X_{n} =1, X_{n+1}=1)\).

  • (5 points) By running repeated simulations estimate \(\lim_{n\ \to \infty} a_n\).
  • (5 points) Compute precisely, \(\lim_{n \to \infty}a_n\). Carefully justify your answer.

6 (5 points)

Let \(\Pi\) be a Poisson point process on the Euclidean plane \(\mathbb{R}^2\). Let \(n \geq 0\) and \(r >0\). Suppose that \(\Pi(r) = \Pi(B(r))\) is the number of points of \(\Pi\) in the disc of radius \(r\) centered at the origin. Identify the conditional distribution of \(\Pi(r)\) given that \(\Pi(2r) = n\).

Endnotes