Estimating unknown parameter exercise.
Hi i started solving exercises for estimating parameters and as i started already i stuck my self with this task:
Estimate the unknonw parameter \theta from the sample : 3,3,3,3,3,7,7,7
drawn from discrete distribution with pmf: P(3) = /theta and P(7)=1-/theta
use two methods : a)moments estimator b) maximum likelihood
-i have some ideas but i kinda dont know how to really apply those, first i thought of taking the first sample moment m1=mean(sample) which is m1=4.2 and make the identity m1=EX (EX is the first moment of the population, the actual discrete distribution that the sample is drawn from) but the problem is i dont know how to express the first moment so i can solve equition for /theta and estimate /theta,
i dont know how the info about the pmf is going to be in use for the problem i dont have idea for that one hmm
for the second method i know that i need to find derivative of the pmf and apply Logarithm and make identity of that to 0 so i can find the extreme points from where ill end up finding the parameter in such manner that the parameter gives the maximum probability that the observed data comes in the same state if i make take sample again but this time with the estimated parameter..
Thanks for helping!