HW 5. Math 504. Spring 2008. CSUF by Nasser Abbasi
problem 3.9 from lecture notes
Part (A)
We first covert the sequence of random variables to sequence of random variables as described. A diagram below will also help illustrate this conversion
We need to show that Using the hint given, we write (for )
Conditioning on , and assuming the pdf of is given by we write
Since the random variables are independent from each others, we break the above 'and' probabilities to products of probabilities.
But , hence the above becomes
But , hence the above becomes
Now do integration by parts (let and
But hence the above becomes
But since it is the integral we started with (see (1)), so move it to the left side, and the above becomes
Hence
is number of records up to time . We need to find and
But and similarly,
Hence
So is a harmonic number. In the limit, this sum is
To find the variance of , we use the hint and assume are independent of each others (i.e. when a record occurs is independent of when previous record occurred), hence the covariance terms drop out (since all zero) and we are left with the sum of variances
But
But
Hence
therefore
Since as gets very large, and as gets very large, then
Part(C)
We need to find where is the time of the first record (not counting which is always a record ofcourse).
Now
Since having no record at and having a record at are indepdent events the above becomes
Similarly,
Similarly,
Hence continuing this way, we see that
Hence
and
Hence