About distribution of random variables

Question

I'm understand what is $\mathrm P$ ($\eta \leq x)$ for a random variable, but what is $\mathrm P$ ($\eta \leq \xi$) if $\xi$ is random variable too? I can't find definition of something like that. How to calculate it, if I know that $\eta, \xi$ are independent and have the same geometric distribution with the parameter $p$. Will it be some function from $p$?

Answer 1

If $\eta$ and $\xi$ are both random variables on the same probability space, then also $\eta-\xi$ is a random variable on that probability space.

Now observe that $$\{\eta\leq\xi\}=\{\eta-\xi\leq0\}$$ so that $$P(\{\eta\leq\xi\})=P(\{\eta-\xi\leq0\})$$

One way to find (mostly not the most convenient one) is finding the distribution of $\eta-\xi$.

Also it can be found as $$\mathbb E[\eta\leq\xi]=\int\int[x\leq y]dF_{\eta,\xi}(x,y)\tag1$$ where $[x\leq y]$ denotes the function $\mathbb R^2\to\mathbb R$ that takes value $1$ if $x\leq y$ and takes value $0$ otherwise, and $F_{\eta,\xi}$ denotes the CDF of random vector $(\eta,\xi)$.

If $\eta$ and $\xi$ are independent then the RHS of $(1)$ becomes:$$\int_{-\infty}^{\infty}\int_{-\infty}^ydF_{\eta}(x)dF_{\xi}(y)=\int_{-\infty}^{\infty}F_{\eta}(y)dF_{\xi}(y)$$