$3.$ Consider the problem of generating sample from a Beta distribution Be$(\backslash$alpha $,\backslash$beta $).$

$($a$)$ One result is$,$ if two Gamma random variables are X$1$ Ga$(\backslash$alpha $,1)$ and X$2$

Ga$(\backslash$beta $,1),$ then

X $=$ X$1$ Be$(\backslash$alpha $,\backslash$beta $).$ X$1+$ X$2$

Use this result to construct an algorithm to generate a Beta random sample. Provide a density histogram to emulate the performance.

$($b$)$ Compare the algorithm in $($a$)$ with the rejection method based on $($i$)$ the uniform distribution; $($ii$)$ the truncated normal distribution.