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Can I say that the random variables are independent by looking at the joint pdf and iff I can factor the two random variables from each other, say (X+1)(Y+2), they are independent?

If not, is there another way one can use to quickly determine based on the joint pdf that two RV's are independent?

Of do I always have to integrate out the other var to find the other marginal pdfs, multiply them together and see if they equal the joint?

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    $\begingroup$ If you have a joint density that factors, you have independence. But one has to be careful, it has to be the whole density function, including where it is $0$. or if you stick to non-zero part, want it over a "rectangle" (possibly infinite) with sides parallel to the axes. $\endgroup$ – André Nicolas Oct 4 '12 at 5:09
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If you can factor the joint pdf $f(x,y)$ into $g(x)h(y)$ then $X$ and $Y$ are indeed independent.

The marginal pdfs are $\dfrac{g(x)}{\int_x g(x) \,dx}$ and $\dfrac{h(y)}{\int_y h(y) \,dy}$ while $\left(\int_x g(x) \,dx\right )\left(\int_y h(y) \,dy\right)=\int_x \int_y g(x)\,h(y)\, dy \,dx=\int_x \int_y f(x,y) \,dy\, dx=1$.

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