# Math Help - Number of records (expected value, variance)

1. ## Number of records (expected value, variance)

Let $X_1, X_2, ... , X_n$ be a sequence of independent identically distributes continous random variables. A record occurs at time j ( $j \leq n$) if $X_j \geq X_i \; \forall \; 1 \leq i \leq j$.
Show that:
$\mathbb{E}$[number of records]= $\sum_{j=1}^{n} \frac{1}{j}$
$\mathbb{D}^2$[number of records]= $\sum_{j=1}^{n} \frac{1-j}{j^2}$

Thank you very much in advance!

2. ## 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.