# How will p-values behave when fitting normal/Poisson to binomial?

I know p-values behave uniformly. Now as p(np) is fixed and n goes to infinity, binomial converges to normal(Poisson). Now suppose I take random binomial samplings and fir normal(Poisson) to it, for say n = 1000. Will my p-value still be uniformly distributed or as binomial converges to normal(Poisson), p-values mostly will be in 0.8-1?

I think what you are referring to is sampling a binomial random variable $$X$$ and then looking at the distribution of $$F(X)$$ where $$F$$ is the binomial CDF. This is actually not uniform; it is a sort of discrete approximation of a uniform variable. In the case where normal approximation becomes valid, it converges to a uniform variable. In the case where Poisson approximation becomes valid, it does not converge to a uniform variable, but rather to the distribution of $$F(P)$$ where $$P$$ has the limiting Poisson distribution and $$F$$ is the limiting Poisson CDF. Again this isn't uniform.
• @AvinashBhawnani Can you explain exactly what you mean by calculating the p-value? I think it is essentially what I am saying here, except that you're plotting $F(X)$ where $X$ are your binomial samples and $F$ is your fitted normal CDF. This will basically look exactly the same as what I said on a histogram with that many data points. – Ian Mar 31 at 18:30