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philipp_w

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Hi there,

I am currently reading Rohatgi's book "An introduction to probabilty and statistics" (http://books.google.de/books?id=IMbVyKoZRh8C&lpg=PP1&hl=de&pg=PA62#v=onepage&q&f=true). My questions concerns the "technique" of finding the PDF of a transformed random varibale Y by a function, let's call it g. So we got Y=g(X), where X is a random variable of continuos type.

There is this method existing, where you first calculate the DF of Y, let us call it F_y, and then obtain the PDF regarding Y by differention of F_y.

There are some special cases, where g is differentiable and absolutely increasing, but the tricky part is, where g does not fullfil such conditions.

My question: In the most presented, more complicated examples in the textbooks, one determines F_y and then differentiates it to obtain the PDF, but what tells me in advance that F_y is in fact an (absolutely) continuous DF? My opinion is, that this technique is more like a guess, to obtain a "possible" PDF.

How can one be sure that F_y is a (absolutely) continuous DF, depending on properties of g, if Y=g(X)? This technique of differentiation, does not work in the general case, am I right? Put it in other words: Of course, one can differentiate F_y, but nothing guarantees in advance that F_y got an integral representation with a continuous function under the integration sign and that this differentiation of F_y results in the PDF, what we are looking for.

How is the correct argumentation or where is my lack of understanding, so that there is no contradiction any longer with the argumentation with the cited textbook from above?

Thanks in advance and best regards

Philipp

I am currently reading Rohatgi's book "An introduction to probabilty and statistics" (http://books.google.de/books?id=IMbVyKoZRh8C&lpg=PP1&hl=de&pg=PA62#v=onepage&q&f=true). My questions concerns the "technique" of finding the PDF of a transformed random varibale Y by a function, let's call it g. So we got Y=g(X), where X is a random variable of continuos type.

There is this method existing, where you first calculate the DF of Y, let us call it F_y, and then obtain the PDF regarding Y by differention of F_y.

There are some special cases, where g is differentiable and absolutely increasing, but the tricky part is, where g does not fullfil such conditions.

My question: In the most presented, more complicated examples in the textbooks, one determines F_y and then differentiates it to obtain the PDF, but what tells me in advance that F_y is in fact an (absolutely) continuous DF? My opinion is, that this technique is more like a guess, to obtain a "possible" PDF.

How can one be sure that F_y is a (absolutely) continuous DF, depending on properties of g, if Y=g(X)? This technique of differentiation, does not work in the general case, am I right? Put it in other words: Of course, one can differentiate F_y, but nothing guarantees in advance that F_y got an integral representation with a continuous function under the integration sign and that this differentiation of F_y results in the PDF, what we are looking for.

How is the correct argumentation or where is my lack of understanding, so that there is no contradiction any longer with the argumentation with the cited textbook from above?

Thanks in advance and best regards

Philipp

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