UMD Probability Qualifying Exams/Aug2009Probability

Problem 1Edit

Let   be i.i.d. random variables with moment generating function   which is finite for all  . Let  .

(a) Prove that

  where

  and  

(b) Prove that

 .

(c) Assume  . Use the result of (b) to establish that   almost surely.


SolutionEdit

(a)  

Thus far, we have not imposed any conditions on  . So the above inequality will hold for all  , hence for the supremum as well, which gives us the desired result.


(b)   where the last equality follows from the fact that the   are independent and identically distributed.

(c)

Problem 2Edit

Let   be a probability space; let   be a random variable with finite second moment and let   be sub  -fields. Prove that

 


SolutionEdit

Problem 3Edit

Let   be independent homogeneous Poisson processes with rates  , respectively. Let   be the time of the first jump for the process   and let   be the random index of the component process that made the first jump. Find the joint distribution of  . In particular, establish that   are independent and that   is exponentially distributed.



SolutionEdit

Show   is exponentially distributedEdit

Let   be the first time that a Poisson process   jumps.

 

  is a Poisson Process with parameter  Edit

Proof: There are three conditions to check:

(i)   almost surely

(ii) For   is   independent of  ? This is true since both   are Poisson Processes and are both independent of each other.

(iii) For   is   distributed Poisson with parameter  ? This is true since the sum of independent Poisson processes are also poison. (see second bullet)

Joint distribution of (J,Z)Edit

 

 

Problem 4Edit

Let   be a martingale sequence and for each   let   be an  -measurable random variable. Define

 

Assuming that   is integrable for each  , show that   is a martingale.


SolutionEdit

Problem 5Edit

Let   be an i.i.d. sequence with   and  . Prove that for any  , the series   converges almost surely.

SolutionEdit

Define  . Then   and  . We check the three components of Kolmogorov's three-series theorem to conclude that   converges almost surely.


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Problem 6Edit

Consider the following process   taking values in  . Assume   is an i.i.d. sequence of positive integer valued random variables and let   be independent of the  . Then

 

SolutionEdit