Good afternoon :). I have a problem I am stuck on in conditional probability, because I am not sure about some steps. I checked for similar posts, but I am actually looking to understand some specific steps of my reasoning, so I hope you won't hold me accountable if a similar problem was once posted.
Ok, so say I have two random variables $X$ And $Y$ following Poisson processes of parameters respectively $\lambda_1$ and $\lambda_2$.
It is quite straightforward to see that the probability that $X$ happens first, for instance, is $\dfrac{\lambda_1}{\lambda_1 +\lambda_2}$.
But now, say we know that at $t = 1$, the event $X$ has happened once.
Problem: What is the expected time at which the first even (of X and Y) happened?
Ok so I decided to split it into two cases. Call $1_X$ the event that $X$ happens first (given that $X$ only happens once in the interval $(0,1)$, and call $1_Y$ the event that $Y$ happens first.
- First case. Assume $1_X$, so $X$ happens first, say at time $t \in (0,1)$. The time to the event follows and exponential distribution with parameter $\lambda_1$,so we could argue that the expectation of $(X|1_X)$ is simply the expectation of this exponential distribution over $(0,1)$, which is $1 - e^{-\lambda_1}$.
But then I started wondering... is that correct? Have we accounted for the fact that no other $X-$event happens in the interval $(t, 1)$? Because it might be the case that $X$ happens another time between $t$ and $1$, which then contradicts our hypothesis that $X$ happened once at $t = 1$.
This is my first question, I will label it Q1 to make my post (and maybe the answers) clearer.
- Second case: Y happens first. We have no condition on the number of times Y has to happen. All we require is that $X$ happens once between $t = 0$ and $t = 1$ (call $t_X$ the time at which $X$ happens).
I made even less progress on that case because I don't know how to treat the fact that we need $Y$ to happen between $t = 0$ and $t = t_X$. If we did not have this condition, then the expected time would simply be $1 - e^{-\lambda_2}$, because we would integrate an exponential random variable over $(0,1)$.
But here, do we integrate the exponential random variable over $(0,t_X)$, and then do another integration over the range of $t_X$?... (This is my second question, I will label it Q2).
I'm sorry if this post is not clear enough. I am quite confused because I never really know, in conditional probability, if something has been accounted for already or not. So I don't really know how to deal with that problem.
Thank you for your answer. Let me know if something needs clarification.
Answer has been posted in the comments, I need to consider the expectation of the minimum of $X$ and $Y$, which eventually comes down to being the expectation of the minimum of a uniform and an exponential random variables.