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Defining a distribution

  1. Apr 22, 2007 #1
    Say if I had a random variable [tex]X[/tex] that followed a Beta distribution [tex]B(a,b)[/tex], and [tex]a[/tex] and [tex]b[/tex] were random variables.

    How would I define the distribution of [tex]X[/tex]??
  2. jcsd
  3. Apr 22, 2007 #2


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    Well, I imagine you would first want to talk about the joint distribution of X, a, and b. Then you can worry about trying to compute the marginal distribution of X.
  4. Apr 22, 2007 #3
    Basically Hurkyl said it depends on how a and b are distributed. Once you know this you can calculate the distribution of X using the standard procedures.
  5. Apr 22, 2007 #4
    Why would X be a marginal distribution??
    Wikipedia says
    But in my situation I do not want to ignore the information from the other variables, ie. variables a and b.

    It seems to me that I should look for the conditional distribution ie. [tex]X|a,b[/tex]??
  6. Apr 29, 2007 #5
    Let [tex]f(x,y)[/tex] be the joint density of [tex]a[/tex] and [tex]b[/tex]. (You can modify the following if one or both of them are discrete.) Then

    [tex]F_X(c) = P(X \le c) = E[P(X \le c | (a,b))].[/tex]

    This gives

    [tex]F_X(c) = \int_0^\infty\int_0^\infty
    P(X \le c | (a,b) = (x,y))f(x,y)\,dx\,dy.[/tex]

    By hypothesis, the distribution of [tex]X[/tex] given [tex](a,b)[/tex] is Beta. So

    [tex]F_X(c) = \int_0^\infty\int_0^\infty

    where [tex]c\mapsto I_c(x,y)[/tex] is the CDF of a Beta random variable with parameters [tex]x[/tex] and [tex]y[/tex].
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