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Probabilities, CDF

  1. Sep 20, 2013 #1
    1. The problem statement, all variables and given/known data

    Given the function F(x) = 1 - e^(-ax) - axe^(-ax) for x>=0 and 0 elsewhere, for which values of 'a' does the function constitute a CDF?

    2. Relevant equations


    3. The attempt at a solution

    I started with saying for that to occur, provided x1>x2 -> F(x1) > F(x2)

    Though I've only managed to say for x1=0 and x2=1 -> a>0 but I'm not sure if it's an unequivocal answer.

    How can I prove it decisively?
     
  2. jcsd
  3. Sep 20, 2013 #2

    mfb

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    Staff: Mentor

    That should be ≥.

    What about F(x)≥0? F(x)≤1?
    Did you check the derivative?

    If all of them are fine, you are done.
     
  4. Sep 20, 2013 #3

    HallsofIvy

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    If you want to show that F is an increasing function, then show that its derivative is never negative.
     
  5. Sep 20, 2013 #4
    Oh, good idea (+thanks). I'll look into it. I may return if I see an issue.
     
  6. Sep 20, 2013 #5
    I've got a bit of trouble. The derivative>0 appears to conclude to (a^2)xe^(-ax)>0 which appears to be true for any 'a' belonging in IR. It appears to be more direct to do 0<=F(x)<=1 which appears to conclude to 0<=(e^-ax)(1-ax)<=1 but how do I learn how to solve that inequality?
     
  7. Sep 20, 2013 #6

    Ray Vickson

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    Aren't you forgetting something very important? If ##f(x) = a^2 x e^{-ax} 1_{\{x \geq 0 \}},## then you need two things:
    [tex](1) \; f(x) \geq 0 \text{ for all }x; \text{ and }\\
    (2)\; \int_{-\infty}^{\infty} f(x) \, dx = 1.[/tex]
    In the current case the integral goes from 0 to ∞ because f(x) = 0 for x < 0. So, if you have a = 0 do these conditions hold? What about if a < 0?
     
  8. Sep 20, 2013 #7
    Well I can I show that for a negative 'a' F will not be approaching 1 for x->+oo. The integral of f(x) between 0 and +oo appears to be 'a' times the expected value of any Exp(a) so it's 1 for any 'a', hrm.. provided a>0 (for any Exp(a)) so I guess that might be enough. It's a bit confusing since the exercise goes on to ask for the fx(x) in a later question so I'm unsure if it needs it exposed directly on this first question.
     
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