13. Let X₁, X₂, ..., X_n be independent and identically distributed continuous random variables. We say that a record value occurs at time j, j ≤ n, if X_j ≥ X_i for all 1 ≤ i ≤ j. Show that

(a) E[number of record values] =

$$

\sum_{j=1}^n 1/j

$$;

(b) Var[number of record values] =

$$

\sum_{j=1}^n \left( 1 - \frac{1}{j} \right)^2

$$