# How to convert the parameters in a binomial distribution to those in a beta distribution?

I know that the beta distribution is the generalized continuous case of the discrete binomial distribution. Let's say I have a binomial distribution, $$B(N,p)$$. I would like to know the corresponding $$\alpha$$ and $$\beta$$ such that the binomial CDF of any count, $$k$$, in $$\{0,1,2,...,N\}$$ equals to the beta CDF at $$x = k/N$$.

In terms of equations, I would like the functions $$f(x)$$ and $$g(x)$$ such that:

1. $$\alpha = f(N,p)$$ and $$\beta = g(N,p)$$, and
2. CDF of $$B(k,n,p) =$$ CDF of $$Beta(k/N, \alpha, \beta)$$ where
• $$N$$ in $$\mathbb{N}$$,
• $$p$$ in $$(0,1)$$, and
• $$k$$ in $$\{0,1,2,...,N\}$$

I have been in search for a while but I find no direct answer. I don't have much maths background and I am hoping for an easy answer in terms of $$f(x)$$ and $$g(x)$$.

• – Tim May 27 '19 at 16:21

I do not know that the beta distribution is the generalized continuous case of the discrete binomial distribution, and I am curious why you think that. $$p^x(1-p)^{n-x}$$ is not really the same as $$x^{\alpha-1}(1-x)^{\beta-1}$$. What I do know is that the beta distribution provides a conjugate family for the parameter $$p$$ of the binomial distribution, but that is a rather different statement

You have a problem in that the cumulative distribution functions are not going to match like that. For example, for $$X$$ binomial you have $$P(X\le 0)=\frac{1}{(1-p)^n}$$ while for $$Y$$ beta-distributed you have $$P\left(Y \le \frac0n\right) =0$$

So if you want an approximate match between the two distributions, an alternative could be to match the means and variances. That could suggest using something like $$\alpha=(n-1)p, \beta=(n-1)(1-p)$$ or in the other direction $$n=\alpha+\beta+1, p= \frac{\alpha}{\alpha+\beta}$$

For example with $$n=4, p=\frac13$$ this might suggest $$\alpha=1, \beta=2$$, and the following chart is what the two cumulative distribution functions would look like; the relationship is not bad, but you would not use one to calculate the other

The simplest relationship between beta and binomial distributions is

cdf_binomial(N,i,p) = sf_beta(p,i+1,N-i) (where sf_beta, the survival function is 1-cdf_beta)


or equivalently, cdf_binomial(N,i,p) = cdf_beta(1-p,N-i,i+1)

where i >= 0 and i < N

• This is a bit short, can you for example explain what is sf_beta? – kjetil b halvorsen May 19 '19 at 13:53