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!