Estimate gamma function using monte carlo Let $\Gamma(\beta) = \int_0^\infty x^{\beta - 1} e^{-x} dx$ how to estimate the above gamma function using monte carlo? Any idea?

Estimating an integral over a non-finite domain using Monte Carlo directly is subtle. Instead, transform variables, say to $u = \frac{1}{1+x}$. The integral becomes

$$\int_0^1 \left( \frac{1-u}{u}\right)^{\beta-1}e^{-\frac{1-u}{u}}\frac{1}{u^2}du $$

Now roll an ensemble of $N$ uniform variates $\\{u_i\\}$ on $(0,1)$, and evaluate $$\frac{1}{N}\sum_{i=1}^N \left( \frac{1-u_i}{u_i}\right)^{\beta-1}e^{-\frac{1-u_i}{u_i}}\frac{1}{u_i^2} $$ Needless to say there are much more efficient ways of doing the integral, but this is true to the basic Monte Carlo idea.