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Math Help - Unbiased estimators

  1. #1
    Senior Member chella182's Avatar
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    Unbiased estimators

    So I'm revising my stats by doing a past exam paper, and I'm stuck on one of the questions about unbiased estimators. The question goes...

    Suppose that X_1, X_2,..., X_n are a random sample from a population with mean \mu and variance \sigma^2.
    a) Prove that the mean estimator \bar{X} is unbiased for \mu. State carefully any formulae you use.
    b) Prove that \bar{X} has variance \frac{\sigma^2}{n}. State carefully any formulae you use.
    It's probably really simple like the last unbiased estimator Q I posted a while back, but I'm just stumped.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by chella182 View Post
    So I'm revising my stats by doing a past exam paper, and I'm stuck on one of the questions about unbiased estimators. The question goes...

    Suppose that X_1, X_2,..., X_n are a random sample from a population with mean \mu and variance \sigma^2.
    a) Prove that the mean estimator \bar{X} is unbiased for \mu. State carefully any formulae you use.
    b) Prove that \bar{X} has variance \frac{\sigma^2}{n}. State carefully any formulae you use.


    It's probably really simple like the last unbiased estimator Q I posted a while back, but I'm just stumped.
    \overline{X}=\frac{1}{n}\sum_{i=1}^n X_i

    By linearity of the expectation operator:

    E(\overline{X})=\frac{1}{n}\sum_{i=1}^n E(X_i)

    but by definition E(X_i)=\mu, so:

    E(\overline{X})=\frac{1}{n}\sum_{i=1}^n \mu=\mu

    Which is the definition of an unbiased estimator for \mu.

    CB
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  3. #3
    Senior Member chella182's Avatar
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    Thank you. Does sound rather simple, I just find it difficult to get my head around this sort of stuff.
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