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Homework Help: Cumulative distribution and density functions

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

    Let X be a random variable with probability density function:

    0.048(5x-x2) IF 0 < x < 5

    0 otherwise

    Find the cumulative distribution function of X

    a) If x ≤ 0, then F(x) =

    b) If 0 < x < 5, then F(x) =

    c) If x ≥ 5, then F(x) =

    2. Relevant equations

    Not quite sure

    3. The attempt at a solution

    The answer to a) is 0. The answer to c) is 1.

    I am making the reasonable assumption that a) is 0 because there is no probability at that point, and that c) is 1 because after that, all probability has been "used" so to speak. However, integrating the function between 0 and 5 does not work. It seems as if my professor totally skipped over teaching us this particular type of problem. Statistics usually makes a good deal of sense to me, but this is pretty foreign.
  2. jcsd
  3. May 1, 2013 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    You say "integrating the function between 0 and 5 does not work". What about it does not work?

    In fact, if we define f(x) = 0 for x < 0 and for x > 5, then the cumulative distribution F(z) is
    [tex] F(z) = \int_{-\infty}^z f(x) \, dx \\
    = 0 \; \text{ if } z < 0,\\
    = \int_0^z (48/1000)(5x - x^2) \, dx \; \text{ if } 0 \leq z \leq 5,\\
    = 1 \; \text{ if } z > 5.[/tex]
    Do the integration to see what you get.

    Are you sure your course notes or textbook do not have any similar examples?
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