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Cumulative distribution function

  1. Jul 14, 2006 #1
    Once again, i'm having a disagreement with my TA regarding a problem set he gave us.

    Here is the exact question, as written:
    Find the distribution function associated with the following density functions:
    a) [tex] f(x) = 3(1-x^2) [/tex] , x an element of (0,1)
    b)[tex] g(x) = x^{-2} [/tex], x an element of positive real numbers

    here's where my problem lies.
    For a)
    if you solve for the cdf, you get:
    [tex] F(x) = \int_{0}^{x} 3(1-t^2) dt = [3t - t^3]_{0}^{x} = 3x - x^3 [/tex] for x element (0,1)

    however, this value of F(x) takes the value of 2 when x = 1, which violates the property of a cdf! Also, if you take the integral of the density from 0 to 1, you will get 2! again a violation of the property of a density function, as the integral should be between 0 and 1.

    for b)
    if you solve for the cdf, you get
    [tex] F(x) = \int_{0}^{x} t^{-2} dt = [-t^{-1}]_{0}^{x} = - 1/x + \infty [/tex] for x element positive real

    which again is greater than 1 for any value of x, positive real.

    and if you get the integral of the density function from 0 to infinity (as the density function is defined for all positive real), you get infinity! which is not between 0 and 1.

    the TA, however said that there is NOTHING wrong with the questions, even after he inspected it.

    Am I insane to think that the questions are wrong? or am I not seeing something obvious?

    help please. I'm going crazy.
     
    Last edited: Jul 14, 2006
  2. jcsd
  3. Jul 15, 2006 #2
    anyone have an idea? thanks.
     
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