Calculating a density=1

Question

I had an excercise regarding a family of densities. My question is: Is this really a family of densities? I calculated the integral over $R^2$ and think that it is only a density when a=1, but I might be wrong. Let $$g(x,y)=\frac{1}{4}e^{-a|x|-a^{-1}|y|}$$ so I calculated the integral over $R^{2}$ and determined that is is one iff a=1, can anyone defy? $a\in(0,\infty)$

Answer 1

$\int_{R^2} g(x,y)dxdy=(\frac{1}{2}\int_R e^{-a^{-1}|y|}dy)(\frac{1}{2}\int_R e^{-a|x|}dx)={(a)(\frac{1}{a})=1}$,

for all $a\gt 0$.