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The observed value of mean of random variable from N($\theta$, 1) distribution is 2.3. If the parameter space is {0,1,2,3} then the maximum likelihood estimate of $\theta$ is?

a) 1

b) 2

c) 2.3

d) 3

I understand that MLE of $\theta$ is $\sum X_i/n$ which is the same as observed value of mean (i.e 2.3). However, it is not in the parameter space.

Intuitively, we have to maximum likelihood

L = $\frac{e^{\sum{(x_i-\theta)^2}}} {\sqrt{2\pi^n}}$

How do I proceed from this?

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    $\begingroup$ calculate L at 0,1,2,3. which is the maximum value? $\endgroup$
    – seanv507
    Commented Jan 23 at 9:58
  • $\begingroup$ Rhea: It's kind of a trick question in that the usual MLE is not part of the parameter space. So, the MLE in this case has to be chosen by checking all the values and taking the max as seanv507 said. By doing that, you are still calculating the MLE because those are the possible values in the parameter space. $\endgroup$
    – mlofton
    Commented Jan 23 at 14:53

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