Understanding the Beta-function

Question

I always forget whether the beta function, B$(\alpha, \beta)$, is defined as $\Gamma(\alpha+\beta)/\Gamma(\alpha)\Gamma(\beta)$ or $\Gamma(\alpha)\Gamma(\beta)/\Gamma(\alpha+\beta)$. Is there an intuitive way, preferably as a normalising constant for the beta distribution, I can understand this function so I never have to look it up again. :)

Answer 1

The beta function is defined in a manner “opposite” to that of binomial coefficients. Since binomial coefficients are defined as $${a+b\choose a}={a+b\choose b}=\frac{(a+b)!}{a!b!}$$ then the beta function is defined as $$B(a,b)=B(b,a)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$$

Answer 2

One must have in mind that he beta density is proportional to the function $x\mapsto x^{\alpha-1}(1-x)^{\beta-1}$ on $(0,1)$. When $\alpha\gt1$ and $\beta\gt1$, this function is $\lt1$ everywhere hence the normalizing constant $\mathrm{B}(\alpha,\beta)$ must be $\lt1$. On the other hand, when $\beta\gt1$ is fixed and $\alpha\to\infty$, the ratio $\Gamma(\alpha+\beta)/(\Gamma(\alpha)\Gamma(\beta))$ is $\gt1$. Thus, the only reasonable option valid for every $\alpha$ and $\beta$ is that $\mathrm{B}(\alpha,\beta)$ is the inverse of this ratio, that is, $\mathrm{B}(\alpha,\beta)=\Gamma(\alpha)\Gamma(\beta)/\Gamma(\alpha+\beta)$.