- #1
psie
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- 28
- Homework Statement
- I guess it is Olympic games, so here's a probability question. In a ##100##-meter Olympic race the running times can be considered to be ##U(9.6,10.0)##-distributed. Suppose there are eight competitors in the finals. What is the probability that all eight finish within the time interval ##(9.8,9.9)##?
- Relevant Equations
- This is in a chapter on order statistics, specifically in a section on the joint density of the extreme order statistics, i.e. ##X_{(n)}=\max\{X_1,\ldots,X_n\}## and ##X_{(1)}=\min\{X_1,\ldots,X_n\}##. Their joint density is $$f_{X_{(1)},X_{(n)}}(x,y)=\begin{cases} n(n-1)(F(y)-F(x))^{n-2}f(y)f(x),&x<y\\ 0&\text{otherwise}.\end{cases}$$ The range is ##R_n=X_{(n)}-X_{(1)}## and it also has a known density which may be of interest (not sure).
I'm not sure how to solve this. Intuitively, we'd want to know $$P(X_{(1)}>9.8,X_{(8)}<9.9),$$but is the above probability simply ##P((X_{(1)},X_{(8)})\in (9.8,10.0)\times(9.6,9.9))## and do I integrate the joint density over ##(9.8,10.0)\times(9.6,9.9)## then?