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Sample Topics

Bayes Estimation

Sample Questions:

1. Consider a statistical decision (e, M,, L) with sample space X where Θ-(01, θ2), H and ye [0,0.25)). Find the minimax decision. in R2 plane. Further show that the class of all non-randomized Bayes decisions.

2. In a point estimation problem e-(0,) A (0, and X follows Poisson distribution with parameter 8 and a sample of size one is made available. Show that the estimator T0X) – X is not Bayes but a generalized Bayes under quadratic loss.

Stochastic Process

Sample Questions: 

Joe Zilch is practicing basketball by repeatedly making attempts (shots) to put the ball in the basket. We label his first shot as random variable (RV) XI , second shot as X2,…, nth shot as X,, etc. When he takes the nth shot, he either makes a basket (X, -1) or misses (X,-0 ). He finds that the result of any shot x, depends on the outcome of his last two shots X -2 and X- as follows:

P(X-1 l he missed both of his last two shots) 1/2

P(X-1 l he made one of his last two shots) 2/3

P(X1 l he made both of his last two shots) – 3/4

a). Show how Joe’s basketball play may be modeled using a Markov chain. How many states are needed? (Hint: Define a state as the outcome of his last two shots). Draw a labeled state transition diagram or trellis describing the process

b). Find the transition matrix P for the process

c). Given that he missed his first two shots, find the joint probability that Joe makes shots number 3 and 4. d). Joe made his first two shots with probability 0.5 and missed his first two shots with probability 0.5. Given these facts, find the joint probability that Joe makes shots number 3 and 4

e). Find the probability that Joe makes any single shot in the long run.

Testing of Hypothesis

Sample Questions:

1.  A recent article in the Arizona Republic indicated that the mean selling price of the homes in the area is more than $220,000. Can we conclude that the mean selling price in the Goodyear, AZ, area is more than $220,000? Use the .01 significance level. What is the p-value?

Probability Distribution

Sample Questions:

1. Let r є N. Let X1,X2, be identically distributed random variables having finite mean m, which are r-dependent, i.e. such that XkXk,.,Xk, are independent whenever kiti > ki +r for each i. (Thus, independent random variables are 0-dependent.) Prove that with probability one, X Xi – m as n -oo. Hint: Break up the sum Ση! Xi into r different sums.

Probability

Sample Questions:

1. Bob, Kevin, and Jack are playing the following game: a single player will toss 5 fair coins. Assume the tosses are independent , and let H = 1 and T 0. Before the coin tosses, without consulting each other, each player chooses some number of the tosses (at least one) and calculates the sum (mod 2) of the tosses. That is the player’s score for example, supposed Kevin chose the second, third and fourth tosses, and Jack chose the first, second, and last toss. If the following coin tossing sequence occurs: HTHHT then Kevin’s score is 0 and Jack’s score is 1. Assume that each player does not select exactly the same subset of tosses.

(a) Show that the scores of the three players are pairwise independent  

(b) Show that the scores of Bob, Kevin, and Jack are not always mutually independent. (A counter example when they are not independent is sufficient, but obviously explain your answer.)