Say we have 2 geometric random variables $X$ and $Y$ with parameters $a$ and $b$ respectively. I am wondering 2 things:
- why is the distribution of $\min(X,Y)$ the same for all geometric random variables with the same parameters?
- When $X$ and $Y$ are dependent, is it still the case that we have uniquely specified the distribution of $X$ and $Y$ in the same way?
For context, a problem I was looking at used the fact that such a distribution is uniquely specified when $X$ and $Y$ were independent, leading me to ask my question.
On an intuitive level, I feel that in both cases the distribution of $\min(X,Y)$ is not uniquely specified- as which sample points correspond to which values in the distributions of $X$ and $Y$ seems relevant to how $\min(X,Y)$ is distributed; and this information isn't specified when we have 2 geometric random variables in general. So what is the reason why the distribution of $\min(X,Y)$ is uniquely determined here?