Question

There are two balls with different qualities in a black box, a red ball and a purple ball. If you randomly pick one ball from this black box and denote the result using x variable, let x=1 if you get a red ball, and x=0 if you get a purple ball, further suppose the probability to get a red ball is α. You play this game of choosing a ball N times independently, i=1,2,3,...N, denote the joint result of this N times game as J. Let x₁, x₂.... x_N be i.i.d observations from this game, xi=1 if red ball with a probability of α; xi=0 if purple ball with a probability of 1-α. You should solve which of the following optimization equations to get the maximum likelihood estimator?

A.

max Pr(J)=Πiα^xi(1-α)^1-xi with respect to α

B.

min Pr(J)=Πiα^1-xi(1-α)^xi with respect to xi

C.

min Pr(J)=∑iα^1-xi(1-α)^xi with respect to xi

D.

max Pr(J)=Πiα^1-xi(1-α)^xi with respect to α

Answer 1