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Homework Help: Probability theory and statistics

  1. Dec 22, 2017 #1
    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. jcsd
  3. Dec 22, 2017 #2

    mfb

    User Avatar
    2017 Award

    Staff: Mentor

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