Is the function 1/|x+y| a joint probability density function?

• Apr 10th 2009, 10:19 AM
crazymonkey
Is the function 1/|x+y| a joint probability density function?
Is the function $\displaystyle \frac{1}{|x+y|}$ a joint probability density function? That is, is $\displaystyle \int\hspace{-6pt}\int_{\mathbb{R}^2} \frac{1}{|x+y|} \ dA$ equal to 1 and is $\displaystyle \frac{1}{|x+y|}$ nonnegative?

This problem is really giving me a headache, been attempting it for a week now and the deadline's today. (Headbang)
• Apr 10th 2009, 10:59 AM
NonCommAlg
Quote:

Originally Posted by crazymonkey

Is the function $\displaystyle \frac{1}{|x+y|}$ a joint probability density function? That is, is $\displaystyle \int\hspace{-6pt}\int_{\mathbb{R}^2} \frac{1}{|x+y|} \ dA$ equal to 1 and is $\displaystyle \frac{1}{|x+y|}$ nonnegative?

This problem is really giving me a headache, been attempting it for a week now and the deadline's today. (Headbang)

no! the integral is divergent: your function is positive wherever is defined. show that the integral is $\displaystyle \infty$ in the first quadrant and thus it's divergent over $\displaystyle \mathbb{R}^2.$