0

I have been reading about generating random samples using direct methods. I have come across a definition that if, $X_i ~ Unif(0,1)$ and are i.i.d., then: \begin{align} Y = \frac{\sum_{i=1}^m logX_i}{\sum_{j=1}^{m+n} logX_j} \end{align}

will be a Beta distribution B(m,n). Why is it so and how would i be able to prove the same?

StubbornAtom
  • 17,932
Abijith P Y
  • 3
  • If you're more interested in results than method, then you can use functions built into various statistical software, e.g. rbeta in R. // Wikipedia has pages discussing various families of distributions, often with a section of efficient simulation. – BruceET Oct 05 '20 at 08:23
  • This is because $-\ln X_i$ has an exponential distribution and $Y=\frac{U}{U+V}$ where $U=-\sum_{i=1}^m \ln X_i$ and $V=-\sum_{i=m+1}^{m+n} \ln X_i$ are independent Gamma variables. – StubbornAtom Oct 05 '20 at 08:44

0 Answers0