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I know that $F(x,y)= \mathbb P [X \leq x, Y \leq y] = \int^x_{-\infty}\int^y_{-\infty}f_{X,Y}(u,v)dvdu$, is differentiated with respect to $x$ and $y$ to calculate the joint density $f_{X,Y}(x,y)$. I have seen in a few places that $\partial^2_{xy}\mathbb P [X \leq x, Y \leq y] = \partial^2_{xy}\mathbb P [X > x, Y \leq y] $, so that either expression can be used to calculate the joint density. I can't seem to verify if this is true or not.

I have the following relation: $\mathbb P [X > x, Y \leq y] + \mathbb P [X \leq x, Y \leq y] = \mathbb P [Y \leq y]$ from the law of total probability, and so calculating just $\partial_x$ of this expression gives:

$\partial_x \mathbb P [X > x, Y \leq y] + \partial_x \mathbb P [X \leq x, Y \leq y] = 0$. And so $\partial^2_{xy} \mathbb P [X > x, Y \leq y] + \partial^2_{xy} \mathbb P [X \leq x, Y \leq y] = 0$, which implies $\partial^2_{xy}\mathbb P [X \leq x, Y \leq y] = -\partial^2_{xy}\mathbb P [X > x, Y \leq y] $.

This seems to say that the densities from either expression gives different results. Does my logic here make sense? Any help would be appreciated. Thanks!

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1 Answer 1

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What you have done is correct. The second formula is wrong.

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