Let $ X \sim \text{Bin}(n,p) $ and $ Y \sim \text{Beta}(r,n-r+1) $. Show , without integration by parts, that $P(X \ge r ) = P (Y \le p)$.
From which point of view I answer this question.
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$\begingroup$ @Xi'an: The question says show the equality without integration by parts. Then how can I show? I don't understand. $\endgroup$– A.DCommented Jan 8, 2015 at 16:47
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$\begingroup$ can you write down any of the two probabilities? $\endgroup$– Xi'anCommented Jan 8, 2015 at 16:48
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$\begingroup$ @Xi'an Yes. $P[X \ge r] = \sum_{x=r}^n {n \choose x} p ^x (1-p)^{n-x}$ $\endgroup$– A.DCommented Jan 8, 2015 at 16:51
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$\begingroup$ Try to do the same with $$\frac{(n+r)!}{(r-1)!(n-r)!}\int_0^p y^{r-1}(1-y)^{n-r} \text{d}y$$ $\endgroup$– Xi'anCommented Jan 8, 2015 at 17:50
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1$\begingroup$ @Xi'an I'm curious where you would go with that integral, given that you are enjoined from integrating by parts. (That injunction probably should be broadly interpreted as an extremely strong hint that no integration at all will be necessary.) Did you have some particularly simple or elegant manipulation in mind? $\endgroup$– whuber ♦Commented Jan 8, 2015 at 18:58
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