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I'm having some trouble with a homework problem, and would appreciate help.

You have an i.i.d. sample of n observations drawn from a distribution.If $\hat x$, the mean, is known, we can estimate the variance with: $\hat \theta_x^2 = 1/n \sum(X-\hat x)^2$. Show that $\hat \theta_x^2$ is and unbiased estimator of the variance $E[(X-\hat x)^2]$.

I understand that the sample variance is unbiased if we divide it by n-1, but why is it also unbiased, according to the problem, if we divide by n?

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    $\begingroup$ I think you must mean that $\hat x$ is the (theoretical) mean, i.e., $\mathbb E X$, not the sample mean $\frac{1}{n} \sum_{i=1}^n X_i$. $\endgroup$
    – cardinal
    Commented Sep 26, 2012 at 1:48
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    $\begingroup$ Think about the wording of your question. The sample mean is always known once you have the sample in hand! :-) But, it obviously cannot be that normalizing by both $n$ and $n-1$ yield unbiased estimators. For further evidence, try what I suggested in the first comment and see if you can prove whether it is true. :-) $\endgroup$
    – cardinal
    Commented Sep 26, 2012 at 2:16
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    $\begingroup$ I edited the question because you sample from a distribution not a random variable and the sample meanis not known in advance. As cardinal said dividing by n leads to an unbiased estimate for the variance when the population mean is given (i.e. known). When you use the sample mean the unbiased estimator is obtained only by dividing by n-1. $\endgroup$
    – Michael R. Chernick
    Commented Sep 26, 2012 at 2:45
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    $\begingroup$ @MichaelChernick: I know your edit was very well-meaning and I appreciate it. However, editing out a potential factual error can drastically change the question meaning and interpretation. My experience is that these errors are often highly correlated with the questioner's actual question and misunderstanding. I believe that may be the case here, hence caution is advised. See also: Should I edit a factual error? $\endgroup$
    – cardinal
    Commented Sep 26, 2012 at 2:58
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    $\begingroup$ @cardinal My thinking is as follows: The question was asked and you gave the appropriate answer. The OP seems to clearly have misquoted the homework problem or the homework problem was written in error. The wording as you pointed out was "the sample mean is known". But the sample mean is never known until it is computed from the data in which case you would normally say that it is estimated. Of course that interpretation is also consistent with dividing by n. This has already been hashed out so why not have the question make logical sense? $\endgroup$
    – Michael R. Chernick
    Commented Sep 26, 2012 at 3:13

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A hint at the answer If X is N(μ, σ$^2$) then E[(X - μ)$^2$] = σ$^2$. So what should you divide by when you add n terms of the form (X$_i$ - μ)$^2$ so that the expectation is still σ$^2$?

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  • $\begingroup$ The question doesn't seem to specify normality anywhere. $\endgroup$
    – Glen_b
    Commented Sep 24, 2019 at 12:18

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