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Math Help - estimators

  1. #1
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    estimators

    A random sample x_{1}...x_{n}, n=200, is taken from a distribution with mean \mu and variance \sigma^2, both are unknown. Find unbiased estimators of the mean and variance, given
    \sum_{i=1}^{200}x_{i}=308 and \sum_{i=1}^{200}x_{i}^2=3784


    I don't even know where to start. Can anyone give me any help please?
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  2. #2
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    Quote Originally Posted by calliope View Post
    A random sample x_{1}...x_{n}, n=200, is taken from a distribution with mean \mu and variance \sigma^2, both are unknown. Find unbiased estimators of the mean and variance, given
    \sum_{i=1}^{200}x_{i}=308 and \sum_{i=1}^{200}x_{i}^2=3784


    I don't even know where to start. Can anyone give me any help please?
    The sample mean is an unbiased estimator of the mean and the sample variance with (n-1) rather than n in the formulation is an unbiased estimator or the variance.

    RonL
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  3. #3
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    Thanks a lot =).

    I got 1.54 as an estimate of the mean and 11.86.. for the variance. I'm hoping that sounds about right.
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  4. #4
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    Quote Originally Posted by calliope View Post
    Thanks a lot =).

    I got 1.54 as an estimate of the mean and 11.86.. for the variance. I'm hoping that sounds about right.
    The mean is correct, but you should have:

    \sigma_{n-1}^2=\frac{n}{n-1}\left( \overline{x^2}-\overline{x}^2 \right) =\frac{200}{199}\left( \frac{3784}{200}-1.54^2 \right)

    for the unbiased estimator of the variance.

    RonL
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  5. #5
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    Yeah, I realised since posting that last message that I made a stupid calculation error. Thanks a lot for your help.
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