UMD Probability Qualifying Exams/Aug2006Probability

Problem 1Edit

Consider a four state Markov chain with state space  , initial state  , and transition probability matrix

 

(a) Compute  .

(b) Let  . Compute  .


SolutionEdit



Problem 2Edit

If   are independent uniformly distributed random variables on [0,1], then let   be the second smallest among these numbers. Find a nonrandom sequence   such that   converges in distribution, and compute the limiting distribution.


SolutionEdit

 

The two terms on the right hand side look like the limit definition of the exponential function. Can we choose   appropriately so that it is?

Let  . Then  

This is the distribution of  .

Problem 3Edit

Suppose that the real-valued random variables   are independent, that   has a bounded density   (for  , with respect to Lebesgue measure), and that   is integer valued.

(a) Prove that   has a density.

(b) Calculate the density of   in the case where   Uniform[0,1] and   Poisson(1).



SolutionEdit

(a)

 

where the last equality follows from Monotone Convergence Theorem.

Hence, we have shown explicitly that   has a density and it is given by  .


(b) When   Uniform[0,1] and   Poisson(1), we have   and   with support on  .

Then from part (a), the density will be

 .

Problem 4Edit

Let   be a Poisson process with unit rate, and let

 

where   is the indicator of the event  .

(a) Find a formula for   in terms of  .

(b) Show that if   with   a fixed constant, then   in probability.


SolutionEdit

(a)Edit

We know that   is distributed as Poisson with parameter  . So

 

Then  


(b)Edit

If   then we must have   only finitely often. The probability of this even (from part a) is  .

This decays to 0 for  . Then clearly, we see that the probability of   is equal to 1.

I don't know how to show the result for  ...

Problem 5Edit

Let   and for   where the r.v.'s   are i.i.d. with  .

(a) Prove that there exist constants   such that   and   are martingales.

(b) If  , then prove that   almost surely and find  .

(c) Prove that   is not a uniformly integrable martingale.

SolutionEdit

(a)Edit

We want  . We can compute both sides of this equation explicitly.

 

Thus if we want this equality to hold we must have  .

Similarly, if we want   then

 

We can easily check that   gives a trivial solution to the equation. Using the substitution   we can find another solution for  . We should get  .

(b)Edit

We've just shown that   is a martingale. Thus,  . Then since each   is i.i.d., we can apply the Strong Law of Large Numbers to say   almost surely. In other words,   almost surely and so certainly   almost surely.

Now to calculate  . We introduce new notation: let  . Then  

by a symmetry argument.

So we can write  . But I don't know how to calculate  ....

(c)Edit

Recall from part (a) that the nontrivial solution for   must be some negative number. Then   almost surely by part (a) as well.

However,  . This by the definition, means the martingale is not right closable. A martingale is right-closable iff uniformly integrable. Thus, we're done.

Problem 6Edit



SolutionEdit