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It is well-known that if $X$ and $Y$ are independent binomial random variables with parameters $(n_1,p)$ and $(n_2,p)$ respectively, $Z = X+Y$ has a binomial distribution with with parameters $(n_1+n_2,p)$.

I was wondering if $Y$ is binomial with parameters $(n_2, 1-p)$ what will be the distribution of $X+Y$?

MikeL
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  • Your first statement is true if $X$ and $Y$ are independent and not if they're not. The answer to the question at the end also depends on information about the joint distribution that you've omitted. If, for example, $Y$ is simply $n_1-X$, then $X+Y$ would be a constant, equal to $n_1$. But if they're independent (the opposite extreme) then the distribution of the sum would be non-trivial. ${}\qquad{}$ – Michael Hardy Aug 18 '14 at 16:07
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    You are right, I should have explicitly mentioned that $X$ and $Y$ are independent. I'll correct the question. – MikeL Aug 18 '14 at 16:09

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