How to identify the limits of triple integral where variables are functionally related?

Question

I have to calculate $P(X > Y+Z, X > Y)$, where $X, Y$ and $Z$ are i.i.d. exponential (1).

So, I know that the joint pdf is just the product of marginal pdfs: $e^{-(x + y + z)}$

I have been given that the probability evaluates to $1/4$, but when I work out the triple integral, I get answers that are either functions of $y$ or $z$.

I guess I am taking the wrong limits for the 3 variables in the triple integral. Currently, I am using

$x$: from 0 to infinity,
$y$: from 0 to $x-z$ (I also tried $y$: from 0 to $x$),
$z$: from 0 to $x-y$

The event for which I need the probability is: $X > Y+Z$ and $X > Y$

I'd be grateful for any advice on where I am going wrong with the limits for the triple integral...

Answer 1

You have to be careful when there are multiple constraints. Observe that $X$ dominates $Y+Z$ and $Y$, so $X$ is free. After considering $X>Y$, we see that $X-Y>Z>0$, which means $Z$ is dominated by $X-Y$. \begin{align} &P(X > Y+Z, X > Y)\\ &=\int_0^\infty\int_0^x\int_0^{x-y}e^{-x-y-z}dzdydx\\ &=\int_0^\infty e^{-x}dx\int_0^x e^{-y}dy\int_0^{x-y}e^{-z}dz\\ &=\int_0^\infty e^{-x}dx\int_0^x(1-e^{y-x})e^{-y}dy\\ &=\int_0^\infty e^{-x}dx\int_0^x(e^{-y}-e^{-x})dy\\ &=\int_0^\infty e^{-x}dx\cdot(1-e^{-x}-xe^{-x})\\ &=\int_0^\infty e^{-x}dx-\int_0^\infty e^{-2x}dx-\int_0^\infty xe^{-2x}dx\\ &=1-\frac12+\frac12\int_0^\infty xd(e^{-2x})\\ &=\frac12+\frac12\left[xe^{-2x}|_0^\infty-\int_0^\infty e^{-2x}dx\right]\\ &=\frac14 \end{align}

Answer 2

In case you aren't compelled to compute the integrals, here is a fun way of solving the problem:

We know that $X, Y, Z \sim \text{Expo}(1)$ independently, and we are interested in $P(X>Y+Z)$.

The intuition is that if $X>Y$ (with half chance), then $X$ has a half chance (again) to be $>Z$.

Formally, $$ \begin{align} P(X>Y+Z) &= P(X>Y)\cdot P(X>Y+Z|X>Y) \quad \text{by conditioning} \\ &= \frac12 \cdot E(P(X>Y+Z | X>Y, Y, Z)) \quad \text{by symmetry and the tower rule} \\ &= \frac12 \cdot E(P(X > Z | Y, Z)) \quad \text{by memoryless property of Expo} \\ &= \frac12 \cdot P(X>Z) \quad \text{by tower rule (in reverse)}\\ &= \frac12 \cdot \frac12 = \frac14 \quad \text{by symmetry} \end{align} $$

This technique easily generalises to arbitrarily many random variables on the right hand side, so if all random variables are i.i.d $\text{Expo}(1)$ then

$$P\left(X > \sum_{i=1}^n Y_i\right) = \frac{1}{2^n}.$$