Suppose X is a discrete random variable with

P(X=0) = 2y/3

P(X=1) = y/3

P(X=2) = 2(1-y)/3

P(X=3) = (1-y)/3

Where 0<=y<=1. The following 10 independent observations were taken from such a distribution: (3,0,2,1,3,2,1,0,2,1).

Find the method of moments estimate of y, an approximate standard error for your estimate, the MLE of y, and an approximate standard error of the MLE.

I have found the method of moments estimate of y (5/12) and the MLE (.5) but I'm not sure how to go about approximating the standard errors. What I initially did for the SE of the first estimate was to calculate the different y's based on the observed probabilities of the X's, then add the squared differences between them and 5/12, divide by 4, and take the squared root, but that doesn't seem quite right. Sorry to ask such an elementary question, but I'm really puzzled as to how to do this. Thanks in advance!

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