# Joint distribution of mid p-value and p-value

I have a question about the joint distribution of the mid p-value and p-value.

We know that, for right tailed test with discrete test statistic $X$ with distribution $F$, the p-value is defined as $P=Pr(X \geq observed~X)$ and the mid p-value is defined as $mid~P=Pr(X \gt observed~X)+\frac{1}{2}*P(X=Observed)$.

I would like to find out $Pr(P_{mid} \leq t~and~P \gt t)$ for some $t \in (0,1)$. Here $P$ is the p-value and $P_{mid}$ is the mid p-value.

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What test do you have in mind? –  whuber Aug 10 '12 at 23:13
Thanks so much for the reply. I am hoping to get a general solution for any discrete distribution F. –  user13154 Aug 11 '12 at 1:06
The answer depends strongly on the test statistic, not just the distribution: that's why you need to tell us what test you're using. –  whuber Aug 11 '12 at 21:05
Ok. We can focus on the Fisher's exact test. Thanks. –  user13154 Aug 12 '12 at 2:31
Does anyone have any thought on it, please? Thanks –  user13154 Aug 14 '12 at 14:01
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