If $X \sim B(n, p)$ and $Y \sim B(m, p)$ are dependent binomial variables with the same probability $p$, and same number of elements $N$, does that make $X + Y$ a binomial variable as well? If so with what parameters?
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No, because the the Binomial distribution is for a series of independent "Bernouilli" trials. If the two variables are dependent then the trials are no longer independent.
To see this intuitively, take an extreme where the two variables are fully dependent, i.e. $X=Y$. Then every time a trial was successful in one variable it would be in the other (and conversely a failure in one would mean a failure in the other), so your variable $X+Y = 2X = 2Y$ would only take even numbered values.
TooTone
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