Fall 2011 Homework 8: Question 3
Athletes compete one at a time at the high jump. Let $X_{j}$ be how high the jth jumper jumped, with $X_{1}, X_{2},...$ with a continuous distribution. We say that the jth jumper set a record if $X_{j}$ is greater than all of $X_{j-1}, ....,X_{1}$.
(a) Find the variance of the number of records among the first n jumpers (as a sum). What happens to the variance as $n\rightarrow \infty$?
(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 $\frac{1}{j(j-1)}=\frac{1}{j-1}-\frac{1}{j}$). What happens to the mean as $n\rightarrow \infty$?
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.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."