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Let $X$ and $Y$ be two discrete uniform i.i.d random variables distributed over $\{0, 1, 2,\ldots, N\}$. Find the pmf of $Z = \min(X, Y)$.

From what I understand, I have to find the joint $pmf$ first, which is just $1/(N+1)^2$ by independence. Now, I have to find the probability function $P(X_{(1)})$. Is this right? If so, how do I determine $P(X_{(1)})$ for $X_i$ discrete?

From two textbooks I have, I only found $P(X_{(1)} = x_i)$. Is $P(X_{(1)})$ = $P(X_{(1)} = x_i)$ for a generic $x_i$?

Thanks.

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$P(\min(X,Y) = k) = \sum_{j=k}^N P(X = k)P(Y = j) + \sum_{j=k}^N P(Y = k)P(X = j) - P(Y=k,X=k) $

$= 2\sum_{j=k}^N P(X = k)P(Y = j)-\frac{1}{(N+1)^2}$ by symmetry.

$ = 2\sum_{j=k}^N \frac{1}{N+1}\frac{1}{N+1} = \frac{2(N+1-k)}{(N+1)^2}-\frac{1}{(N+1)^2}$

$ = \frac{2(N+\frac{1}{2}-k)}{(N+1)^2}$

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  • $\begingroup$ Would you mind posting the formula that gets you this result? $\endgroup$ – Victor S. May 12 at 22:46
  • $\begingroup$ Intuitively, if you want $\min(X,Y)=k$, then one of them must equal $k$, and the other can be anything above $k$, thus you get the above sum which is a sum over all valid combinations of $(X,Y)$ in the sample space $\endgroup$ – George Dewhirst May 12 at 22:52
  • $\begingroup$ I wanted to notify you that your answer is not correct as you can verify that your probability function does not sum up to 1. For example, fix N = 3, sum k from 0 to 3. Then you get 10/8. Besides, the pmf I was looking for is the pmf of the order statistic $X_{(1)}$. If you’re interested, check Casella’s Statistical Inference, 2nd edition, pages 226-228, specially equation 5.4.3. $\endgroup$ – Victor S. May 14 at 4:12
  • $\begingroup$ Yes should be $\frac{2(N+\frac{1}{2}-k)}{(N+1)^2}$ as you count the diagonal term $X=k,Y=k$ twice in summation. $\endgroup$ – George Dewhirst May 14 at 11:18
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$P(Z\le n)=P(X\le n\ or\ Y\le n)=P(X\le n)+P(Y\le n)-P(X\le n \ and\ Y\le n) =\frac{2(n+!)}{N+1}-(\frac{n+1}{N+1})^2$

The last step uses the independence of $X$ and $Y$.

Since the random variables are discrete I am not sure what you would want for a density function.

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