TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Levajkovic)/Übungen 2023W/HW04.4

The winning result

Assume that the times of fifteen 100-metres sprinters are independent random variables with common ${\mathcal {N}}(10,0.1252)$-distribution. Compute the probability that the winning results is below

9.75, i.e. compute the probability that the minimum of their running times is less than 9.75.

Dieses Beispiel ist als solved markiert. Ist dies falsch oder ungenau? Aktualisiere den Lösungsstatus (Details: Vorlage:Beispiel)

Lösungsvorschlag von Lessi

--Lessi 2024-02-07T13:04:11Z

Let $X_{1},\dots ,X_{15}$ be the times of 15 sprinters running 100m, which are iid. random variables with $X_{i}\sim {\mathcal {N}}(10,0.125^{2})$.

The question is, whats the probability that the minimum is less than 9.75s. We want to compute the probability that at least one of the athletes runs faster/less than 9.75s. So let Y be the number of athletes running less than 9.75, therefore we want the probability $P(Y\geq 1)=1-P(Y=0)$ The event $Y=0$ happens exactly when all athletes run slower than 9.75, therefore its probability is:

$\prod _{1\leq i\leq 15}P(X_{i}\geq 9.75)=(1-P(X\leq 9.75))^{15}$

So finally the probability that the minimum (e.g at least one of them) is faster than 9.75s is $1-(1-P(X\leq 9.75))^{15}=0.29$

1 - pnorm(9.75, 10, 0.125, lower.tail = FALSE)^15