I am struggling with the following question:
Consider a random variable (RV) Y that follows the distribution:
$$P(Y=y|p) = (1-p)^yp$$
where $y \in$ {0,1,2,..} is a non-negative integer, and p is the parameter of the distribution. Imagine we observe a sample of n non-negative integers y={$y_{1},....y_n$) and want to model them using the given distribution ( the data is independently and identically distributed).
Write down the likelihood function for the data y ( i.e, the joint probability of the data under the given distribution with the probability parameter p )
I have got as far as the following ( I think) but am lost on the next step.
$\prod_{i=0}^y p \prod_{i=0}^y(1-p)^y$
I have tagged maximum-likelihood because there is no likelihood tag.
The likelihood function of the parameter $p$ given the data $(y_1,\cdots,y_n)$ is $$ L(p\mid y)=P(Y_1=y_1,\cdots, Y_n=y_n\mid p)=\prod_{i=1}^n P(Y_i=y_i\mid p) $$ where $Y_i$'s are iid copies of $Y$.
Now you simply do the substitution.