Marginal density using lebesgue integral

Asked Jan 03 '18 at 13:24

Active Jan 07 '18 at 06:49

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If we have two random variables X and Y with a joint density function $f_{XY}(x,y)$. How can I express the marginal density functions $f_{X}(x)$ and $f_{Y}(y)$ using the Lebesgue integral? I know that

$f_{X}(x) = \int^\infty_{-\infty}f_{XY}(x,y)dy$
$f_{Y}(y) = \int^\infty_{-\infty}f_{XY}(x,y)dx$

but how can I express these integrations using the Lebesgue integration without using the Lebesgue measure and without using the change of variable formula?

Thanks

edited Jan 07 '18 at 06:49

asked Jan 03 '18 at 13:24

user 2000

But this is already Lebesgue integration... – Olivier Oloa Jan 03 '18 at 13:25
I am sorry, I am confused , how is this Lebesgue integration? don't we need to integrate with respect to a certain measure? – user 2000 Jan 03 '18 at 13:27
Here it's the Lebesgue measure on the real line. – Olivier Oloa Jan 03 '18 at 13:29
I am sorry, I still don't get it. because I always look at how we find the expectation, for example $E[X] = \int x f_{X}(x) dx$ but using Lebesgue integral $E[X] = \int X dP$ , so in the Lebesgue integral $f_{X}(x)$ and $dx$ are not used. So I how can I apply the same principle to the marginal density? – user 2000 Jan 03 '18 at 13:34
Ok, @Jasmine I leave it to an expert in probability :) – Olivier Oloa Jan 03 '18 at 13:37
@Jasmine Maybe this helps – Fakemistake Jan 07 '18 at 08:21
@Fakemistake I want to write the integration with respect to $dP$, something similar to the way we write the expectation – user 2000 Jan 07 '18 at 17:21

Marginal density using lebesgue integral

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