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I'm really stuck on this problem and I have no clue how to proceed right now.

Let $X,Y$ be random geometric variables. Let $Z = min(X,Y)$, and $W = { \left\{ \begin{array}{lll} 0 & \mbox{if } X<Y \\ 1 & \mbox{if } X=Y \\ 2 & \mbox{if } X>Y \end{array} \right. }$

Find the joint pdf of $Z$ and $W$.

I'm not really sure where to go with this problem. Any suggestions for where to start would be greatly appreciated.

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  • $\begingroup$ First you think about the values that $(Z,W)$ take on. For instance does when$X<Y$? what is $(Z,W)$? etc. Then compute the probabilities of those values. By the way no density at play here since you are working with discrete RVs, its all PMFs and a joint PMF. $\endgroup$ – Nap D. Lover Jan 23 at 1:03
  • $\begingroup$ It could also be useful to know how the minimum of two (presumably identically and independently distributed) geometric RVs is distributed... maybe start there too. Its been asked on this site many times. $\endgroup$ – Nap D. Lover Jan 23 at 1:05
  • $\begingroup$ @LoveTooNap29 I can see that when $X < Y, (Z,W) = (X, 0)$, but I'm not quite sure how to turn this into a joint pmf. I'm trying to do this without the assumption that $Z$ and $W$ are independent, so I don't want to just multiply the pmf of $X$ by the probability that $X < Y$. How would I go about finding the pmf for $(Z,W) = (X,0)$ without using independence? $\endgroup$ – rocketPowered Jan 23 at 2:10

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