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

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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)

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

=$\int _{-\infty }^{x}\left(\int _{-\infty }^{y}f_{X,Y}(x,y)dy\right)dx$

=$\int _{-\infty }^{x}\left(\int _{-\infty }^{y}f_{X}(x)f_{Y}(y)dy\right)dx$

=$\int _{-\infty }^{x}f_{X}(x)\left(\int _{-\infty }^{y}f_{Y}(y)dy\right)dx$

=$\left(\int _{-\infty }^{x}f_{X}(x)dx\right)\left(\int _{-\infty }^{y}f_{Y}(y)dy\right)$

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

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

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