This is exercise 2.2.1 from Achim Klenke: »Probability Theory — A Comprehensive Course«

Let $X$ and $Y$ be independent random variables with $X \sim \exp_\theta$ and $Y \sim \exp_\rho$ for certain $\theta,\rho > 0$. Show that $$\mathbf{P}[X < Y] = \frac{\theta}{\theta +\rho}\, .$$

Now, in practice, this exercise is easy. $\exp_\theta$-distribution is defined as $$ \mathbf{P}[X \leq x] = \int_0^x \theta e^{-\theta t} \, dt \quad \text{ for } x \geq 0\, .$$

We just have to evaluate the integral: $$\int_0^\infty \mathbf{P}[X \leq x] \cdot \rho e^{-\rho x} \, d x = \int_0^\infty \Bigl(\int_0^x \theta e^{- \theta t} \, d t \Bigr) \cdot \rho e^{-\rho x} \, d x\, ,$$

which gives $\frac{\theta}{\theta +\rho}$.

But how does one do it rigorously?

Why is the following possible: $$\mathbf{P}[X < Y] = \int_0^\infty \mathbf{P}[X \leq x]\cdot \mathbf{P}[Y = x] \, d x \\ \text{ and using } \mathbf{P}[Y = x] = \rho e^{-\rho x} \, ?$$

Convolution of real valued random variables hasn't been defined yet.

  • 4
    $\begingroup$ You are not doing any convolutions. Because $X$ and $Y$ are independent, the conditional probability that $X \leq y$ given that $Y = y$ is the same as the unconditional probability $P\{X \leq y\}$. Thus, your calculation is a use of the law of total probability: $$P\{X < Y\} = \int_0^\infty P\{X \leq y \mid Y = y\}f_Y(y)\,\mathrm dy = \int_0^\infty P\{X \leq y\}f_Y(y)\,\mathrm dy$$ $\endgroup$ – Dilip Sarwate Jan 18 '15 at 4:56
  • $\begingroup$ @DilipSarwate the problem is that law of total probability or conditional probability were not introduced at this time in the book, so the integral is completely unjustified at this point of the book $\endgroup$ – Masacroso May 21 at 8:16

The most closest answer to this exercise that I had found, that fit to the context, is assuming that we already knows, at the time of the exercise, the theory of Lebesgue.

Then, as stated prior to the exercise of the book, we already knows that the joint distribution of two independent random variables $X$ and $Y$ with densities $f_X$ and $f_Y$ is defined by

$$\begin{align}\Pr[X\le x, Y\le y]&=F_{XY}(x,y)\\&=\int_0^x\int_0^y f_X(t)f_Y(s)\, ds\, dt\\&=\int_{[0,x]\times[0,y]}f_X(t)f_Y(s)\, d(s,t)\end{align}\tag1$$

Thus because $\Pr$ is a measure from $(1)$ we have that

$$\Pr[(X,Y)\in A]=\int_A f_X(t)f_Y(s) d(t,s),\quad A\in\mathcal L([0,\infty)^2)\tag2$$

Hence choosing $A:=\{(x,y)\in [0,\infty)^2: x<y\}$ we find that

$$\begin{align}\Pr[(X,Y)\in A]&=\int_0^\infty\int_0^t f_X(s)f_Y(t)\,ds\, dt\\ &=\int_0^\infty F_X(t) f_Y(t)\, dt\\&=\int_0^\infty(1-e^{-\theta t})\rho e^{-\rho t}dt\\ &=1-\rho\int_0^\infty e^{-(\theta+\rho)t}dt\\&=1-\frac{\rho}{\rho+\theta}=\frac{\theta}{\theta+\rho}\end{align}\tag3$$

However I dont found an answer that fit well to the theory of the book when this exercise was presented. Probably some variation of this reasoning could be justified in the context where the exercise appear.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.