1. Xander tosses a fair 6-sided die with outcomes 1, 2, 3, 4, 5, 6 each equally likely. Let X denote the random outcome of this die roll. As soon as Xander tosses the die, Yolanda calculates Y = 7 − X.
(a) What is the probability that X ≤ 3 and Y ≤ 4?
(b) Are X and Y independent? Why or why not?
(c) Are X and Y correlated? Why or why not?
(d) Which is greater, Var(X) + Var(Y ) or Var(X + Y )?
(e) Once the random values of X and Y have been recorded, Bob decides to run a standard Brownian motion B(t) and compare the values of this stochastic process at the random times t = X and t = Y . What is the probability that B(X) < B(Y )?
2. Let Xn, n = 0, 1, 2, 3, . . . be the simple random walk in the graph below.
Figure 1: A simple graph on 4 vertices {a, b, c, d} with 4 edges.
The 1-step transition probabilities defining Xn are
Pij = 1 deg i if i → j connected by an edge 0 otherwise
where deg(i) is the number of edges coming out of the vertex i.
(a) Determine the 1-step transition matrix P.
(b) If X0 = a, what is the probability that X4 = a?
(c) If X0 is chosen uniformly in {a, b, c, d}, what is the probability that X1 = b?
(d) Find the equilibrium distribution π of Xn.
(e) If X0 = c, what is the probability that Xn returns to c infinitely-many times?
3. Let N (t) be a Poisson process with rate λ = 1 and arrival times S1, S2, . . ..
(a) State the properties characterizing the Poisson process.
(b) What is the probability that S2 − S1 < S3 − S2?
(c) What is the probability that N (1) = 2 and N (3) = 5?
(d) What is the probability that S1 ≤ 1.4 and S2 ≤ 2.4?
(e) What is lim Var N(t) ?
4. Consider the triangular probability distribution 0 elsewhere
where θ > 0 is an unknown parameter. Let (Y1, . . . , Yn) be a random sample from this distribution.
(a) Calculate the mean µ and variance σ2 of this distribution.
(b) Find an estimator θˆ1 of θ by the method of moments.
(c) Is θˆ1 unbiased? Is it consistent?
(d) Let θˆ2 be the maximum likelihood estimator of θ. Show that θˆ2 ≥ max1≤i≤n |Yi|.
(e) Assuming max1≤i≤n |Yi| > 0, show that θˆ2 is uniquely determined by
ni = n.i=1 θˆ2 − |Yi|
That is, show that x = θˆ2 is the only solution to the equation Σn |Yi| = n such that x ≥ max1≤i≤n |Yi|.
(f) Consider the estimator θˆ3 = c Σn i=1 x−|Yi| |Yi|, where c is a real constant. How
should c be chosen for θˆ3 to be unbiased?
(g) From here onwards, it is assumed that c is such that Show that θˆ3 is a consistent estimator of θ.
(h) Prove that θˆ3 is unbiased. √n θˆ3 − θ −→ N (0, θ2/2) as n → ∞,
where N (m, v) is a normal distribution with mean m and variance v.
(i) Deduce an approximate confidence interval of the form [L, ) for θ at the level 1 α. That is, give a formula for L = L(θˆ3, n, α) where α (0, 1). Hint: first find an asymptotically pivotal quantity for θ.
(j) Suppose that you want to test the hypothesis H0 : θ = 1 versus Ha : θ > 1 at the significance level α = 0.01 and observe the following result: θˆ3 = 1.05 with n = 50. Would you retain or reject H0? Hint: use the
asymptotically pivotal quantity of part (i) to conduct the test.
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