2

Let X, Y, Z be independent continuous random variables with exponential density functions

$\lambda e^{-\lambda x}$, $\mu e^{-\mu y }$ and $\nu e^{-\nu x}$ respectively, on $[0,\infty)$ (and zero otherwise)

Find $P(X<Y<Z)$

To be honest I dont not really know where to start with this question. I know that $\lambda e^{-\lambda x}$ is the poisson formula and i know the conditions for independence. For this reason i have not attempted the question as i have been completely clueless about it since i saw it.

Any help much appreciated. Many thanks

Bernard.Mathews
  • 834

2 Answers2

3

Hint: What you want to find is hat you need to calculate is $$ P(X<Y<Z) = \iiint_{\{(x,y,z):x<y<z\}} f_{X,Y,Z}(x,y,z)\,dx\,dy\,dz $$ Since $X,Y$, and $Z$ are independent, the joint density is the product of the individual densities: $f_{X,Y,Z}(x,y,z) = f_X(x) f_Y(y) f_Z(z)$. So, you really only need to calculate $$ P(X<Y<Z) = \iiint_{\{(x,y,z):x<y<z\}} f_X(x) f_Y(y) f_Z(z)\,dx\,dy\,dz $$ it remains for you to find the bounds on the integrals for the region $\{x<y<z\}$ in $\Bbb{R}^3$ remembering that the densities for $X, Y,$ and $Z$ are only non-zero when $0<x$, $0<y$, and $0<z$.

Tom
  • 9,978
2

$$\lambda\mu\nu \int_0^\infty \int_x^\infty \int_y^\infty e^{-\lambda x -\mu y -\nu z}\,dz\,dy\,dx $$

ncmathsadist
  • 49,383
  • 1
    Considering that OP indicated to be in need of aid regarding the concept of joint probability distribution, this can at best be helpful as a comment. – Lord_Farin Nov 27 '13 at 22:02
  • 1
    @Lord_Farin Solution posted before said comment. – Did Nov 27 '13 at 22:44