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

= August 15 =

Independence of two random variables
From Independence (probability theory):


 * 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)