integral support for a density function

Question

$$f_{XYZ}(xyz)=ke^{-(x+y+z)}$$ $$ 0<x<y<z $$ I must find for which k this is a density function. $$\int_{Rx}\int_{Ry}\int_{Rz} ke^{-(x+y+z)} dzdydx =1$$ $$k\int_{Rx}\int_{Ry}\int_{Rz} e^{-x}e^{-y}e^{-z} dzdydx =1$$ But I can't find which are supports of single variable. I tried: $$\int_0^y\int_0^z\int_0^\infty$$ But for this exercise I don't have solutions, so please help me. This is a very stupid question, I know, I'm sorry. Thanks.

Answer 1

You know that $x$ can take any value in $(0,\infty)$; given $x$, $y$ can take any value in $(x,\infty)$; and given $x$ and $y$, $z$ can take any value in $(y,\infty)$. This suggests that you must compute $$ \int_0^\infty\int_x^{\infty}\int_{y}^{\infty}ke^{-x}e^{-y}e^{-z}\,dz\,dy\,dx=k\int_0^{\infty}e^{-x}\left[\int_x^{\infty}e^{-y}\left[\int_{y}^{\infty}e^{-z}\,dz\right]\,dy\right]\,dx. $$

Answer 2

Hint:

Try: $$k\int_{0}^{\infty}\int_{0}^{z}\int_{0}^{y}e^{-\left(x+y+z\right)}dxdydz$$