If we assume the underlying probability of a discrete random variable y is binomial, we have:
---the possible values of y are the (n+1) integer values 0,1,2,...,n
---(nCy*theta^y) = n! / [y!(n-y)!]
1. That statistical estimation problem concerns how to use n and y to obtain an estimator of theta, "theta_hat", which is a random variable since it is a function of the random variable, y.
2. The likelihood function gives the probability of the observed data (i.e., y) as a mathematical function of the unknown parameter, theta.
3. The mathematical problem addressed by maximum likelihood estimation is to determine the value of theta, "theta_hat", which maximizes L(theta)
--The maximum liklihood estimator of theta is a numerical value that agrees most closely with the observed data in a sense of providing the largest possible value for the probability L(theta).
Using calculus to maximize the function,(nCy*theta^y)(1-theta)^(n-y), by setting the derivative of L(theta) with respect to theta equal to zero and then solving the resulting equation for theta to obtain theta_hat: