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]).(adsbygoogle = window.adsbygoogle || []).push({});

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 theminimalcondition 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?

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# Minimum conditions for defining joint PDF

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