(adsbygoogle = window.adsbygoogle || []).push({}); 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

f_{X}= (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 :

F_{X}(t) = P(X≤t) = ∫ (125-x)/450 * dx for x between 95 and t and we have

(250t - t^{2}-14725)/900

3. The attempt at a solution

The right solution is :

n=8 and from X_{1}to X_{n},we can describe independent stochastic variables, the time when the last one came into target:

Z=max(X_{1}....X_{n})

And the time the first hit the finish line:

Y= min(X_{1}....X_{n})

Further:

F_{z}(t) = P(Z<t)= P(max(X_{1},...X_{n})<t)= F_{X1}(t)....F_{Xn}(t)

in the same way:

F_{Y}(t)=1-(1-F_{x1}(t))...(1-F_{Xn}(t))

All in goal after 100 minutes: Z≤100

P(Z<100) = [F_{X}(100)]^{8}= (11/36)^{8}

None in goal after 100 minutes: Y> 100

P(Y>100) = 1-FY(100)= (1-F_{x}(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 0.69^{8}= 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?!

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# Homework Help: Probability theory and statistics

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