Suppose that $X$ is a discrete random variable with $P(X = 1) = p$ and $P(X = 2) = 1-p$. Three independent observations of $X$ are made: $x_1 =2, x_2 = 1, x_3 = 2$
a.) Write out likelihood as a function of $p$.
b.) Find the maximum likelihood estimator.
So I first recognized it as a Bernoulli distribution, and got the likelihood function = $3p(1-p)^2$
Then I derived it and found the max which was $p=1/3$. I'm not exactly sure if this approach was correct, can anyone help? Also, if I wanted to check if this was unbiased and consistent, how would I approach doing this? Thanks.