Let X1, . . . , Xn be a random sample from a normal distribution with mean µ and variance 1. It is known that µ ∈ (0, 1] ∪ [2, 3). Prove that the MLE of µ, if it exists, is a biased estimator of µ.

Ok, so all I know about biasedness is that for an estimate to be unbiased its expected value should be equal to the true value of the parameter. Now, here since µ lies within an interval, the first question that arises in my mind is that should the expected value of µ be taken only wrt those values of (X1, ..., Xn) for which the MLE exists. And if so, how do we proceed further to actually prove that the expected value of MLE is not equal to µ. I have thought about this a lot but have no idea.


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