Testwiki:Reference desk/Archives/Mathematics/2017 August 15

{| width = "100%"

|- ! colspan="3" align="center" | Mathematics desk |- ! width="20%" align="left" | < August 14 ! width="25%" align="center"|<< Jul | August | Sep >> ! width="20%" align="right" |Current desk > |}

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

August 15

Independence of two random variables

From Independence (probability theory)#Two random variables:

X and Y with cumulative distribution functions $F_X(x)$ and $F_Y(y)$, and probability densities $f_X(x)$ and $f_Y(y)$, are independent iff the combined random variable (X, Y) has a joint cumulative distribution function

$F_{X,Y}(x,y)=F_X(x)F_Y(y),$

or equivalently, if the joint density exists,

$f_{X,Y}(x,y)=f_X(x)f_Y(y).$

How can one show that these are equivalent? Loraof (talk) 03:41, 15 August 2017 (UTC)

$F_{X,Y}(x,y)$

=$\underset{-\mathrm{\infty }}{\overset{x}{\int }}(\underset{-\mathrm{\infty }}{\overset{y}{\int }}{f}_{X,Y}(x,y)dy)dx$

=$\underset{-\mathrm{\infty }}{\overset{x}{\int }}(\underset{-\mathrm{\infty }}{\overset{y}{\int }}{f}_X(x)f_Y(y)dy)dx$

=$\underset{-\mathrm{\infty }}{\overset{x}{\int }}{f}_X(x)(\underset{-\mathrm{\infty }}{\overset{y}{\int }}{f}_Y(y)dy)dx$

=$(\underset{-\mathrm{\infty }}{\overset{x}{\int }}{f}_X(x)dx)(\underset{-\mathrm{\infty }}{\overset{y}{\int }}{f}_Y(y)dy)$

=$F_X(x)F_Y(y)$

Bo Jacoby (talk) 11:19, 15 August 2017 (UTC).

Thanks, Bo! Loraof (talk) 16:48, 15 August 2017 (UTC)