Probabilities Dealing with Independent Exponential RVs

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June 28th 2010, 09:31 PM
Chris L T521

Probabilities Dealing with Independent Exponential RVs

This has been bothering me for a while now:

Q: If $X_i,\,i=1,2,3$ are independent exponential random variables with rates $\lambda_i,\,i=1,2,3$ , find

(a) $\mathbb{P}(X_1<X_2<X_3)$

Here's what I figured out so far:

Let $Y=\min(X_2,X_3)$ Then $Y$ is $\mathcal{E}(\lambda_2+\lambda_3)$ and is independent of $X_1$ . Now observe that $\mathbb{P}(X_1<Y)=\frac{\lambda_1}{\lambda_1+\lamb da_2+\lambda_3}$ . We now see that

$\begin{aligned}\mathbb{P}(X_1<X_2<X_3) &= \mathbb{P}(X_1=\min(X_1,X_2,X_3))\mathbb{P}(X_2<X_ 3\mid X_1=\min(X_1,X_2,X_3))\\&=\frac{\lambda_1}{\lambda _1+\lambda_2+\lambda_3}\mathbb{P}(X_2<X_3\mid X_1=\min(X_1,X_2,X_3))\end{aligned}$

Now, take note that the distribution of $X_2-x$ and $X_3-x$ given that they are larger than $X_1 = x$ is the same as that of $X_2$ and $X_3$ , so

$\begin{aligned}\mathbb{P}(X_2<X_3\mid\min(X_2,X_3) \geq x) &= \mathbb{P}((X_2-x<X_3-x)\mid\min(X_2,X_3)\geq x)\\ &=\mathbb{P}(X_2<X_3)\\ &=\frac{\lambda_2}{\lambda_2+\lambda_3}\end{aligne d}$

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At this stage, I'm not sure how to tie it all together to get $\mathbb{P}(X_1<X_2<X_3)$ . Is it valid for me to state that we should simply get $\mathbb{P}(X_1<X_2<X_3)=\left(\frac{\lambda_1}{\la mbda_1+\lambda_2+\lambda_3}\right)\left(\frac{\lam bda_2}{\lambda_2+\lambda_3}\right)$ ?

I would appreciate any help with this! (Nod)
June 29th 2010, 02:32 AM
SpringFan25

I cant comment on your solution but i think there might be an easier way

Spoiler:

you can get the joint pdf easily enough, the probability you want is then

$\displaystyle \intop_{x_1=0}^{x_1=x_2} ~ \intop_{x_2=x_1}^{x_2=x_3} ~ \intop_{x_3=x_2}^{x_3=\infty}f(x1,x2,x3) dx_3 dx_2 dx_1$

I haven't done the integration but i would have thought that it was tractable, given that f() is going to be very simple? If nothing else it could verify your answer
June 30th 2010, 09:41 AM
Moo

Quote:

Originally Posted by Chris L T521

$\begin{aligned}\mathbb{P}(X_2<X_3\mid\min(X_2,X_3) \geq x) &= \mathbb{P}((X_2-x<X_3-x)\mid\min(X_2,X_3)\geq x)\\ &=\mathbb{P}(X_2<X_3)\\ &=\frac{\lambda_2}{\lambda_2+\lambda_3}\end{aligne d}$

Are you sure of the second equality ?

Anyway, if you really want to do something along the lines of what you've already done, can't you write it this way :

$P(X_1<X_2<X_3)=P(X_2<X_3)P(X_1<\min(X_2,X_3)\mid X_2<X_3)$

But $P(X_1<\min(X_2,X_3)\mid X_2<X_3)=P(X_1<X_2)$

So finally, $P(X_1<X_2<X_3)=P(X_2<X_3)P(X_1<X_2)$

Did it go wrong somewhere ? oO

Springfan25 : there's a problem with your boundaries, since for example $x_3$ appears after having integrated with respect to $x_3$ , same for $x_2$ .
June 30th 2010, 10:32 AM
SpringFan25

Quote:

Originally Posted by Moo

But $P(X_1<\min(X_2,X_3)\mid X_2<X_3)=P(X_1<X_2)$

Are you sure about that one?

it must be true that
$\displaystyle (1)~~~~~~P(X_1< X_2) =P(X_1<X_2 | X_2 < X_3)P(X_2 < X_3) + P(X_1 < X_2 | X_2 > X_3)P(X_2 > X_3)$
$\displaystyle (1.1)~~~P(X_1< X_2) =P(X_1<X_2 | X_2 < X_3)P(X_2 < X_3) + P(X_1 < X_2 | X_2 > X_3)(1-P(X_2 < X_3))$

it must also be true that

$\displaystyle (2) ~~~P(X_1<\min(X_2,X_3)\mid X_2<X_3) = P(X_1 < X_2 | X_2 < X_3)$

You propose
$\displaystyle (3)~~~ P(X_1<X_2)=P(X_1<\min(X_2,X_3)\mid X_2<X_3)$

(2) and (3) Together imply:
$\displaystyle (4)~~~P(X_1<X_2) = P(X_1 < X_2 | X_2 < X_3)$

But i think (4) contradicts (1.1)
$\displaystyle (1.1) ~~~P(X_1< X_2) =P(X_1<X_2 | X_2 < X_3)P(X_2 < X_3) + P(X_1 < X_2 | X_2 > X_3)(1-P(X_2 < X_3))$
$\displaystyle (4)~~~~~P(X_1<X_2) = P(X_1 < X_2 | X_2 < X_3)$

I can only see 2 ways that (4) could be consistent with (1.1)
either $\displaystyle P(X_2 < X_3) =1$ (not true in general)

or $\displaystyle P(X_1<X_2 | X_2 < X_3) = P(X_1 < X_2 | X_2 > X_3)$ . This is clearly not the case for any given value of x3 (eg, consider x3=0), and therefore (i think?!) not true....but im not sure about that reasoning.

In the unlikely event that i am right about the above, i think the integration approach could still work. I may have the limits wrong but there is a constructable region R (albeit unbounded) on which the integral needs to be done. Here's my second go:

Spoiler:

$\displaystyle \intop_{x_1=0}^{x_1=\infty} \intop_{x_2=x_1}^{x_2=\infty} \intop_{x_3=x_2}^{x_3=\infty}f_{x_1,x_2,x_3} dx_3 dx_2 dx_1<br />$

Im not sure about those upper limits though. I think it would work for a finite upper limit that was the same for x1, x2, x3; but i dont know if it works with infinity.