# Probability P(X<Y<Z)

• Jun 23rd 2013, 05:51 AM
MagisterMan
Probability P(X<Y<Z)
I'm wondering about the following problem: Let $\displaystyle$f_{X,Y,Z}(x,y,z)=e^{-(x+y+z)}$$for x,y,z>0. Find \displaystyle P(X<Y<Z)$$. I've seen the solution to this at http://www-rohan.sdsu.edu/~babailey/.../lecture10.pdf, but I don't understand how they went from P(X<Y<Z) to the triple integral. How does one work with an expression as P(X<Y<Z)? Thanks.
• Jun 23rd 2013, 08:14 AM
Plato
Re: Probability P(X<Y<Z)
Quote:

Originally Posted by MagisterMan
I'm wondering about the following problem: Let $\displaystyle f_{X,Y,Z}(x,y,z)=e^{-(x+y+z)}$ for x,y,z>0. Find $\displaystyle$P(X<Y<Z). but I don't understand how they went from P(X<Y<Z) to the triple integral. How does one work with an expression as $\displaystyle P(X<Y<Z)$

$\displaystyle P(X<Y<Z)$ mean what it says: points $\displaystyle \{(x,y,z) : 0<x<y<z\}$ so the integrals are $\displaystyle \int_o^\infty {\int_0^z {\int_0^y f } } dxdydz$
• Jun 23rd 2013, 12:55 PM
MagisterMan
Re: Probability P(X<Y<Z)
Thanks. I think I need to brush up on my multivariable calculus... :)