How can I convert a Beta Distribution to a Gamma Distribution?
Strictly speaking, I want to transform parameters of a Beta Distribution to parameters of the corresponding Gamma Distribution. I have mean value, alpha and beta parameters of a Beta Distribution and I want to transform them to those of a Gamma Distribution.
[Math] Transformation of beta distribution into gamma distribution
probability distributionsstatistics
Related Solutions
Once you know that the normalizing factor of the density of the beta distribution with parameters $(a,b)$ is $1/B(a,b)$, you know without calculus that the moments of a random variable $X$ with this distribution are $\mathrm E(X^s)=B(a+s,b)/B(a,b)$ and, more generally, $\mathrm E(X^s(1-X)^t)=B(a+s,b+t)/B(a,b)$.
The rest is here.
The following seems to be a typical definition of pdf for a basic probability course.
I got it from this course site.
You will want to show that the function you have satisfies these conditions.
Best Answer
Let $X_a$ and $X_b$ denote independent gamma random variables with respective parameters $(a,c)$ and $(b,c)$, for some nonzero $c$. Then $\dfrac{X_a}{X_a+X_b}$ is a beta random variable with parameter $(a,b)$.