Could someone help me with setting up the function of this question. I've been setting it up with the gamma distribution function but kept getting the wrong answer.
What I did was I used the Gamma Distribution function to evaluate the integral from 0 to 1, but the answer uses the Gamma Function only, I am confused about this and don't know why the gamma distribution function is NOT used in this case.
Here is the distribution function I used:
And Here is the answer:
Note:
Best Answer
The reason why you're not getting helpful responses is because you need to show your work, rather than simply stating that you did the work. In order to advise you, we need to see what you are doing. If you do not know how to typeset your work, then you need to learn how to do it.
In the solution, the gamma distribution is used. When $\alpha = 2$ and $\beta = 1$, $\Gamma(\alpha) = \Gamma(2) = 1! = 1$. $\beta^\alpha = 1^2 = 1$. $x^{\alpha-1} = x^{2-1} = x$. $e^{-x/\beta} = e^{-x/1} = e^{-x}.$ Therefore, the gamma density becomes $xe^{-x}$ as shown.
But since we can't see your work, how are we supposed to know where you might have had a misunderstanding? That would be the most helpful thing, ideally.
In general, for positive integers $n$, $\Gamma(n) = (n-1)!$.