I'm trying to calculate the likelihood function (not log-likelihood) for a random sample of n observations from a Gamma Distribution and a Weibull Distribution and I'm struggling to work with the pi notation as it's not something I've done much before.
So for the Gamma distribution this is what I have:
$$ \prod_{i=1}^n \frac{x_{i}^{a-1}e^{-bx_{i}}b^a}{\Gamma(a)} =\Gamma(a)^{-n}b^{an}e^{-b\sum x_i}\prod_{i=1}^n x_{i}^{a-1} \\ = \Gamma(a)^{-n}b^{an}e^{-bn \bar{x}}\prod_{i=1}^n x_{i}^{a-1} $$
Is this valid? I just feel uncomfortable with this and feel unsure, also is there a way for me to simplify the remaining product term at all?
For the Weibull, I'm struggling a bit more, this is all I have:
$$ \prod_{i=1}^n \gamma \lambda x_{i}^{\gamma -1}e^{-\lambda x_{i}^{\gamma}} = \gamma^{n} \lambda^{n}e^{-\lambda \sum x_{i}^{\gamma}}\prod_{i=1}^n x_{i}^{\gamma -1} $$
I can't seem to figure out a way to simplify the remaining product and sum of powers of $x_i$
Any help would be appreciated, even some signposting to a resource that explains how to work with products like this as I'm really struggling to find anything.