So I got this exercise from a book and I'm confused by a statement they made.
Example: In a 100-meter Olympic race, the running times can be considered to be $U$~$(9.6, 10.0)$-distributed. Suppose that there are eight competitors in the finals. We wish to find the probability that the winner breaks the world record of 9.69 seconds. All units are seconds.
Solution
We wish to determine $P(X(1) < 9.69)$.
Since $f_X(x) = 2.5$ for $9.6 < x < 10.0$ and zero otherwise, it follows that in
the interval $9.8 < x < 10.2$ we have $F_X(x) = 2.5x − 24$ and hence $F_{X_{(1)}} (x) = 1 − (25 − 2.5x)^8$
(since we assume that the running times are independent). Since the desired
probability equals $F_{X_{(1)}}(9.69)$, the answer to our problem is $1 − (25 − 2.5·9.69)^8 \approx 0.8699.$
My question is that I don't see why to look for the CDF on $9.8 < x < 10.2$ since that interval hasn't been mentioned in the question
It is a typo: they meant $9.6 < x < 10.0$. The CDF is still $F_X(x) = \frac{x - 9.6}{10.0-9.6} = 2.5 x - 24$.