# 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?

## 1 Answer

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.

• Actually what I did was, I took some random 100 random samples from binomial n=1000,p=0.1. Then fit Normal to it, calculate the chi square value and hence the p-value. Ran this 100 times and look at the histogram of 100 p-values. It was a bit uniform looking. So, I was a bit confused as that it should converge to Normal, so p-value should be >0.8 as the fit should be good. But why was it so uniform? – Avinash Bhawnani Mar 31 at 17:41
• @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
• By p-value I meant, the area of the right side in the chi square distribution graph, with the value of chi square found by test. – Avinash Bhawnani Mar 31 at 18:46