HW 5. Math 504. Spring 2008. CSUF by Nasser Abbasi

problem 3.9 from lecture notes

Problem


problem_statment.png

Solution

Part (A)

We first covert the sequence of random variables $X_{n}$ to sequence of random variables $I_{i}$ as described. A diagram below will also help illustrate this conversion


record.png

We need to show that MATH Using the hint given, we write (for $i\geq2$)

MATH

Conditioning on $X_{i}$, and assuming the pdf of $X$ is given by $f\left( x\right) $ we write

MATH

Since the $X_{j}$ random variables are independent from each others, we break the above 'and' probabilities to products of probabilities.

MATH

But MATH, hence the above becomes

MATH

But MATH, hence the above becomes

MATH

Now do integration by parts (let MATH and MATH

MATH

But MATH hence the above becomes

MATH

But MATH since it is the integral we started with (see (1)), so move it to the left side, and the above becomes

MATH

Hence

MATH


Part(B)

$N_{n}$ is number of records up to time $n$. We need to find MATH and MATH

MATH

But MATH and similarly, MATH

HenceMATH

So MATH is a harmonic number. In the limit, this sum is

infinity
. Hence number of records is infinite. i.e. if we wait long enough, we will always obtain a new record.

To find the variance of $N_{n}$, we use the hint and assume $I_{i}$ 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

MATH

But MATH

But MATH

Hence

MATH
, therefore MATH

therefore MATH

Since MATH as $n$ gets very large, and MATH as $n$ gets very large, then

MATH as $n$ gets very large.


Part(C)

We need to find MATH where $T$ is the time of the first record (not counting $n=1$ which is always a record ofcourse). MATH

Now

MATH

Since having no record at $T=2$ and having a record at $T=3$ are indepdent events the above becomes

MATH

Similarly,

MATH

Similarly,

MATH

Hence continuing this way, we see that

MATH

Hence MATH

and MATH

Hence MATH