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Minimum conditions for defining joint PDF

  1. Feb 23, 2008 #1
    Suppose one knows a probability density [itex]p(x)[/itex] over a space [itex]X[/itex] (where [itex]x\in X[/itex]) and a conditional probability density [itex]p(y|x)[/itex] over a space [itex]Y[/itex] (where [itex]y\in Y[/itex]).

    This implies the integral [itex]\int{p(x)dx}[/itex] is well defined as well as [itex]\int{p(y|x)dy}[/itex].

    Defining a joint probability density

    [tex]p(x,y)\ =\ p(y|x)p(x)[/tex] ,​

    [itex]p(x)[/itex] will clearly be its marginal density over [itex]X[/itex], and the double integral

    [tex]P(C)\ = \ \iint\limits_{(x,y)\in C}{p(x,y)dydx}[/tex]​

    is well defined for all measurable subsets [itex]C\ \subseteq \ X\times Y[/itex].


    One commonly assumes that integrals with reversed order of integration are equivalent

    [tex]P(C)\ = \ \iint\limits_{(x,y)\in C}{p(x,y)dxdy}\qquad ,[/tex] ​

    which also implies that the marginal probability density over [itex]Y[/itex] exists and is uniquely defined [itex]p(y)\ = \ \int{p(x,y)dx}[/itex].


    This is not necessarily the case, since the change in order of integration poses restrictions on the integrand [itex]p(x,y)[/itex]. One sufficient condition, I believe, is that [itex]p(x,y)[/itex] is continuous, but that is clearly not the minimal condition required. For instance, change of order of integration can be done for a [itex]p(x,y)[/itex] that involves step functions, which are obviously not continuous.

    Does anyone know what the minimal conditions are for changing the order of integration, in this case?
     
    Last edited: Feb 23, 2008
  2. jcsd
  3. Feb 24, 2008 #2

    gel

    User Avatar

    If p>=0 then you just need to know that p is measurable. More generally, if p is measurable then
    [tex]
    \int\int |p(x,y)|dxdy=\int\int |p(x,y)| dydx
    [/tex]
    and, as long as this is finite then you can commute the integrals in [itex]\int\int p(x,y)dxdy[/itex]. See Fubini's Theorem.
     
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