Athletes compete one at a time at the high jump. Let be how high the jth jumper jumped, with with a continuous distribution. We say that the jth jumper set a record if is greater than all of .

(a) Find the variance of the number of records among the first n jumpers (as a sum). What happens to the variance as ?

(b) A double record occurs at time j if both the jth and (j-1)st jumpers set records. Find the mean number of double records among the first n jumpers (simplify fully; it may help to note that ). What happens to the mean as ?

(a) Find the variance of the number of records among the first n jumpers (as a sum). What happens to the variance as ?

(b) A double record occurs at time j if both the jth and (j-1)st jumpers set records. Find the mean number of double records among the first n jumpers (simplify fully; it may help to note that ). What happens to the mean as ?

Solution:
(a) Var = summation of (1/j - 1/j^2) from j = 1 to n. This goes to infinity as n goes to infinity. Use indicator r.v.s, symmetry, definition of independence (and pairwise independence) and correlation, to arrive at solution.
(b) mean = 1 - 1/n, which goes to 1 as n goes to infinity. Also note the independence of the record indicators.

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