I have two questions.
1) In maximum likelihood estimation of binomial distribution I did as follows :
$L(p)=$$\prod$$ n\choose x_i $$p^{xi}(1-p)^{n-x_i}$
=$ n\choose x_1 $$n\choose x_2 $...$ n\choose x_n $ $p^{\sum x_i}(1-p)^{(n-x_1)+(n-x_2)+...(n-x_n)}$.
My question isn't $(1-p)^{(n-x_1)+(n-x_2)+...(n-x_n)}=(1-p)^{n^2-\sum x_i}$.
Then the M.L.E I get is =$\bar x\over n$ .
Is this wrong?
The article here says $\hat{p}=\frac{\sum_{i=1}^{n}{x}_{i}}{n}=\frac{k}{n}$.
Here why is it written as $\frac{\sum_{i=1}^{n}{x}_{i}}{n}$ and not as $\bar x$. ?
2)Here does ${\sum_{i=1}^{n}{x}_{i}}$ refer to the number of successes in n independent trials?
What is meant by number of successes in n independent trials?
> y<-rbinom(20,20,0.4)
> y
[1] 11 3 5 8 10 9 6 11 7 9 8 5 6 6 6 9 8 10 4 7
> x<-sum(y)
> x
[1] 148
Does it mean if I generate some random numbers as then rbinom gives the number of successes in each trial so
${\sum_{i=1}^{n}{x}_{i}}$ refers to sum of rbinom values. Is the way I have understood ${\sum_{i=1}^{n}{x}_{i}}$ correct ?