# Probability theory and statistics

1. Dec 22, 2017

### Pouyan

1. The problem statement, all variables and given/known data
The time (minute) that it takes for a terrain runner to get around a runway is a random variable X with the tightness function
fX = (125-x)/450 , 95≤x≤125

How big is the probability of eight different runners, whose times are independent after 100 minutes:

a) Everyone has scored?
b) nobody has scored?

2. Relevant equations
I do know the distribution function is :

FX(t) = P(X≤t) = ∫ (125-x)/450 * dx for x between 95 and t and we have
(250t - t2 -14725)/900

3. The attempt at a solution

The right solution is :

n=8 and from X1 to Xn, we can describe independent stochastic variables, the time when the last one came into target:
Z=max(X1....Xn)

And the time the first hit the finish line:
Y= min(X1....Xn)
Further:
Fz(t) = P(Z<t)= P(max(X1,...Xn)<t)= FX1(t)....FXn(t)

in the same way:
FY(t)=1-(1-Fx1(t))...(1-FXn(t))
All in goal after 100 minutes: Z≤100
P(Z<100) = [FX(100)]8 = (11/36) 8

None in goal after 100 minutes: Y> 100
P(Y>100) = 1-FY(100)= (1-Fx(100))8 = (25/36)8

How do I think :
I interpreted "The time everyone has scored after 100 minutes" as P (X> 100)
And tried with P(100<X<125) = 1-Fx(100) = 0.69
Then 0.698 = 0.051
I know it's wrong, but this sentence "After 100 minutes ..." made me dizzy!

Why do we have to use max and min in this case?!

2. Dec 22, 2017

### Staff: Mentor

The two approaches look identical to me and the answers agree apart from rounding errors.