So this is a question from the Rice textbook on mathematical statistics. It is as follows:
Let $T_1$, $T_2$ be independent Exponential random variables with parameters $\alpha$ and $\beta$. Calculate the probability that $T_1 < T_2$.
So I am a bit confused as to how I should start this question. After some research online and consultation with friends, I was told to use the joint distribution of the two functions (which is easy to calculate as the R.Vs are independent and thus we can just multiply their respective probability distributions). However, after this a wacky integral is set up, usually of the form:
$$\int_{0}^{\infty}\int_{0}^{y} f(x,y)dx dy$$ What does this mean? Intuitively speaking, is this the integral of the joint density under the line $y = x$? But these are random variables, ($T_1$,$T_2$), so how would I set up the integral in this case? I am just very confused overall with regard to joint distributions and am not entirely sure where to start.
Best Answer
$$\color{blue}{\int_0^\infty} \color{red}{\int_0^y} f_{T_1, T_2}(x,y) \,\color{red}{dx}\,\color{blue}{dy}$$
$f(x,y)$ is the density function, upon integrating them, we recover the corresponding probability of the interested event.
The integration domain describe the region of interest.
Here $x$ correponds to $T_1$ and $y$ correponds to $T_2$.
The domain of integration does describe the set $\{(x,y)|y \geq x, y\geq 0 , x\geq 0\}$.
First, $y$ can take any nonnegative value. Upon fixing $y$, $x$ can only take value from $0$ up to $y$.