Number of records (expected value, variance)

Let $\displaystyle X_1, X_2, ... , X_n$ be a sequence of independent identically distributes continous random variables. A record occurs at time j ($\displaystyle j \leq n$) if $\displaystyle X_j \geq X_i \; \forall \; 1 \leq i \leq j$.

Show that:

$\displaystyle \mathbb{E}$[number of records]=$\displaystyle \sum_{j=1}^{n} \frac{1}{j}$

$\displaystyle \mathbb{D}^2$[number of records]=$\displaystyle \sum_{j=1}^{n} \frac{1-j}{j^2}$

Thank you very much in advance!

Re: Number of records (expected value, variance)

First af all, you should calculate:

a) P(X_n is a record)=?

then:

b) E [number of records by time n]=?

c) Var [number of records by time n]=?

I give you some hints how to solve these problems:

For part (a) use symmetry and observe that P(X_i = X_j ) = 0

since X_i’s are continuous random variables. For parts (b) and (c) introduce indicator random variables. Note that these indicator random variables are independent

in this case.

I hope I helped you. :)

Re: Number of records (expected value, variance)

Thank you for your help!

On the other hand, to tell the truth, I still can't solve the problem.

I would really appreciate any help!