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I have the distribution of X with respect to parameter t vaying between 0 and 1. However, in nature, parameter t is not uniformly distributed. It has a known probability distribution. What is distribution of X given distribution of t? The product? Can you clarify, and hint at notions or concepts involved?

EDIT: in fact, problem can be formulated this way: X is a function of t,

$X: t -> X(t)$

and I would like to have an expression for P(X(t)) when I known the distribution for t.


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How can the distributions of $t$ and $X$ be independent when the distribution of $X$ depends on $t$? – Tunococ Sep 5 '12 at 9:15
I mean that distribution for parameter t is independent of distribution for X ^^ – antitrust Sep 5 '12 at 9:21
That still doesn't make sense. Independence is symmetric. – Tunococ Sep 5 '12 at 9:26
Thats right, its not independent. X's distribution is a function of parameter t (which is known). And t has a known probability distribution. Question remains: what is then distribution of X knowing distribution of t? – antitrust Sep 5 '12 at 9:34
The distribution of $X$ could be anything. As Tunoco rightly points out, it is in general dependent upon $t$. But special cases could have no such dependency. Take $X:t \mapsto 1$ for instance. – Raskolnikov Sep 5 '12 at 10:12
up vote 1 down vote accepted

Let t have density f and let x=g(t). If g is 1-1 the t=g$^-$$^1$(x). Substitute g$^-$$^1$(x) for t in f and then multiply by the Jacobian of the transformation to get the density for x.

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