An urn contains N coins among which theta of them are perfec

An urn contains N coins among which theta of them are perfectly balanced, (the probabilities for heads and tails is 0.5), while the remaining N - theta coins are two headed. A coin is drawn at random, it is flipped twice, the result is noted, and the coin is returned to the urn without being otherwise inspected. If n repetitions of this experiment yielded 0,1,2 heads respectively, with frequencies n_0; n_1; and n_2, estimate theta using

(a) the method of moments,

(b) the method of maximum likelihood.

Solution

Theta coins have prob for head = 0.5

N-theta have prob for head = 1

A coin is drawn at random.

The coin flipped twice has outcomes either hh with prob 0.5theta +n-theta/n = n-0.5 theta/n

or ht or th for the remaining.