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Thread: 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 $\displaystyle X_1$, $\displaystyle X_2$,..., $\displaystyle X_n$ are a random sample from a population with mean $\displaystyle \mu$ and variance $\displaystyle \sigma^2$.
    a) Prove that the mean estimator $\displaystyle \bar{X}$ is unbiased for $\displaystyle \mu$. State carefully any formulae you use.
    b) Prove that $\displaystyle \bar{X}$ has variance $\displaystyle \frac{\sigma^2}{n}$. 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 $\displaystyle X_1$, $\displaystyle X_2$,..., $\displaystyle X_n$ are a random sample from a population with mean $\displaystyle \mu$ and variance $\displaystyle \sigma^2$.
    a) Prove that the mean estimator $\displaystyle \bar{X}$ is unbiased for $\displaystyle \mu$. State carefully any formulae you use.
    b) Prove that $\displaystyle \bar{X}$ has variance $\displaystyle \frac{\sigma^2}{n}$. 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.
    $\displaystyle \overline{X}=\frac{1}{n}\sum_{i=1}^n X_i$

    By linearity of the expectation operator:

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

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

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

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

    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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