UMD Probability Qualifying Exams/Jan2011Probability

Problem 1Edit

A person plays an infinite sequence of games. He wins the  th game with probability  , independently of the other games.

(i) Prove that for any  , the probability is one that the player will accumulate   dollars if he gets a dollar each time he wins two games in a row.

(ii) Does the claim in part (i) hold true if the player gets a dollar only if he wins three games in a row? Prove or disprove it.


SolutionEdit

(i): Define the person's game as the infinite sequence   where each   equals either 1 (corresponding to a win) or 0 (corresponding to a loss).

Define the random variable   by

  that is,   counts how many times the player received two consecutive wins in his first   games. Thus, the player will win   dollars in the first   games. Clearly,   is measurable. Moreover, we can compute the expectation:

 

Now observe what happens as we send  :


 


Hence the expected winnings of the infinite game is also infinite. This implies that the player will surpass $  in winnings almost surely.

(ii): Define everything as before except this time  

Then   which gives   Thus we cannot assert that the probability of surpassing any given winnings will equal 1.



Problem 2Edit

There are 10 coins in a bag. Five of them are normal coins, one coin has two heads and four coins have two tails. You pull one coin out, look at one of its sides and see that it is a tail. What is the probability that it is a normal coin?


SolutionEdit

This is just a direct application of Bayes' theorem. Let   denote the event that you pulled a normal coin. Let   denote the even that you have a tail.

By Bayes,

 

The probability of seeing a tail on a normal coin,   is 5/20 since there are five tails on normal coins out of all 20 faces. The probability of seeing a tail is 13 out of 20 (5 normal + 2*4 double).

Problem 3Edit

Let   be a Markov chain with state space  , with transition probabilities   for  ,   for  .

(i) Find a strictly monotonically decreasing non-negative function   such that   is a supermartingale.

(ii) Prove that for each initial distribution  



SolutionEdit

(i) Let   be the Markov transition matrix. I claim that for any initial probability distribution,  , then  .

Proof of claim: It is sufficient to consider the case where the initial distribution is singular, i.e.  . Clearly we can see that  . Then   if   and for   we have  .

Now let  . We want to compute   for  .

  where the last inequality comes from our claim above. This shows that   is a supermartingale.

Problem 4Edit

Let   be i.i.d. random variables with  .

(i) Prove that the series   converges with probability one.

(ii) Prove that the distribution of   is singular, i.e., concentrated on a set of Lebesgue measure zero.


SolutionEdit

(i) Notice that

  So the series is bounded. Moreover, it must be Cauchy. Indeed for any   we can select   sufficiently large so that for every  ,   Hence, the series   converges almost surely.


(ii) To show that   is supported on a set of Lebesgue measure zero, first recall some facts about the Cantor set.

The Cantor set   is the set of all   with ternary expansion   (in base 3). This corresponds to the usual Cantor set which can be thought of the perfect symmetric set with contraction 1/3.

Instead, consider the set   consisting of all   with expansion   in base  . There exists an obvious bijection between the elements of   and  . Since the Lebesgue measure of   is  . Hence   has support on a set of Lebesgue measure zero.

Problem 5Edit

Let   be a sequence of independent random variables with   uniformly distributed on  . Find   and   such that   converges in distribution to a nondegenerate limit and identify the limit.

SolutionEdit

This is a direct appliction of Central Limit Theorem, Lindeberg Condition.

We know that each random variable   has mean   and variance  .

Then   and  . Then   converges in distribution to the standard normal provided the Lindeberg condition holds.

Hence we want to check  

Since   grows faster than   then for sufficiently large  , the domain of each integral is empty. Hence the above equation goes to 0 as  . Thus the Lindeberg condition is satisfied and CLT holds.

Problem 6Edit

(i) Let   be random variables defined on a probability space  . Assuming that   for all  , prove that   implies  , i.e. under the above assumptions, almost sure convergence implies convergence in mean square.

(ii) Let  be a random process with the property that   and   are finite and do not depend on   (such a process is called wide-sense stationary). Prove that the correlation function   is continuous if the trajectories of   are continuous.

SolutionEdit

(i) Let  . By assumption  . Now we compute the   norm:

 

Let us evaluate the first integral on the right-hand side. We can write  

  by Fatou's lemma

  (since  ).


Now the second term:

  by the triangle inequality.

  since   all have finite second moments.

Thus we have just shown that under the above assumptions, almost sure convergence implies convergence in mean square.


(ii)