Question

Normalizing a 3D probability distribution

Answer 1

Let X, Y, and Z have the joint probability density function:\[f(x,y,z)= kxy^2z\]Where 0<x, y<1, 0<z<2 , and \[f(x,y,z)=0\]elsewhere. I need to find k such that it's a valid probability distribution function. Here's what I tried:\[k^{-1} = \lim_{a \rightarrow -\infty} \lim_{b \rightarrow \infty} \int\limits_{0}^{2}\int\limits_{a}^{1}\int\limits_{0}^{b}(xy^2z) dx dy dz = \lim_{a \rightarrow -\infty} \lim_{b \rightarrow \infty} \frac{b^2(1-a^3)}{3}\]But this is divergent. What am I doing wrong?