UMD Probability Qualifying Exams/Aug2010Probability

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Problem 1[edit]

Two persons, A and B, are playing a game. If A winsa a round, he gets $4 from B and wins the next round with probability 0.7. If A loses the round, he pays $5 to B and wins the next round with probability 0.5.

(i) Write downt he transition matrix of the Markov chain with two states, {A won the current round, B won the current round} and find the stationary probabilities of the states

(ii) Find .

Solution[edit]

(i) The Markov transition matrix will be the 2x2 matrix where corresponds to a win for Player A and corresponds to a loss for Player A. For example, is the probability that Player A wins after winning in the previous hand; is the probability that Player A wins after losing in the previous hand; etc. This will give


The stationary distribution will be the tuple such that . We can calculate this explicitly:

yields the following system of equations: Using the fact that must be a probability (i.e. ) we get .

(ii) Since is positive, and hence ergodic, then any initial probability distribution will converge to the stationary distribution just calculated, . Thus as Player A will win with probability . Can Player A expect to have more money though? For sufficiently large we can compute Player A's expected winnings in one round:

Thus Player A should expect to have more money than before the game with probability 1.




Problem 2[edit]

(i) Let be a random variable with zero mean and finite variance . Show that for any

.

(ii) Let be a square-integrable martingale with . Show that for any

.

Solution[edit]

(i) were the second-to-last inequality is the standard Chebyshev's inequality.



Problem 3[edit]


Solution[edit]

Problem 4[edit]

Let be random variables with finite expectations.

(i) Show that implies .

(ii) Show that if is identically distributed with , then

Solution[edit]

(i) Let . Easy to see that is convex.

Then by Jensen's Inequality we have

. Taking the expectation on both sides gives

.