UMD Probability Qualifying Exams/Jan2006Probability

Problem 1Edit

Let   be i.i.d. r.v.'s such that   and   a.s., and let  .

(a) Find a number   such that   is a martingale and justify the martingale property.

(b) Define  . Compute  .

(c) Compute  .


SolutionEdit

(a)Edit

Each   is clearly  -measureable and finite a.s. (Hence  ). Therefore we only need to verify the martingale property. That is, we want to show  

 

We can assert that   exists and is finite since each   almost surely. Therefore, in order to make   a martingale, we must have  .



Problem 2Edit

Let   be independent Poisson processes with respective parameters  , where   is an unspecified positive real number. For each  , let  . Show that   does not depend on   and find   explicitly.


SolutionEdit

First let us find the distribution of  :

 

Thus by the chain rule, our random variable   has probability density function

 

So then

 

Now integrate the remaining integral by parts letting  . We get:

 

Repeat integration by parts another   times and we get

 

Problem 3Edit


Let   be independent random variables such that

 

(a) Find the characteristic function of  .

(b) Show that   converges in distribution to a non-degenerate random variable.


SolutionEdit

(a)Edit

 

Then by independence, we have  

Problem 4Edit



SolutionEdit

Problem 5Edit


SolutionEdit

Problem 6Edit



SolutionEdit