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?
Best Answer
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 $\left(\int_x g(x) \,dx\right )\left(\int_y h(y) \,dy\right)=\int_x \int_y g(x)\,h(y)\, dy \,dx=\int_x \int_y f(x,y) \,dy\, dx=1$.