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Math Help - Continuous random variables

  1. #1
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    Continuous random variables

    Im stuck with the following -

    (Xn-1 + Xn)/2;
    (1/4)X1 + (3/4)X2;
    (X1 + X2 + + Xn)/(n 1)

    I need to derive expressions for the expected values and variances of the (three) above estimators of mx.

    Any help greatly appreciated.
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  2. #2
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    Quote Originally Posted by bill2010 View Post
    Im stuck with the following -

    (Xn-1 + Xn)/2;
    (1/4)X1 + (3/4)X2;
    (X1 + X2 + + Xn)/(n 1)

    I need to derive expressions for the expected values and variances of the (three) above estimators of mx.

    Any help greatly appreciated.
    I think we need more information to get any useful answer. Are these sequence of random variables i.i.d?
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  3. #3
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    X is a continuous random variable, and n is the sample size taken from the population.
    It is trying to derive the mean and variance, i have to state which is the best and worst estimator out of the 3 given.

    That is all the information in the question im afriad.
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  4. #4
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    Quote Originally Posted by bill2010 View Post
    Im stuck with the following -

    (Xn-1 + Xn)/2;
    (1/4)X1 + (3/4)X2;
    (X1 + X2 + ……… + Xn)/(n – 1)

    I need to derive expressions for the expected values and variances of the (three) above estimators of mx.

    Any help greatly appreciated.
    Quote Originally Posted by bill2010 View Post
    X is a continuous random variable, and n is the sample size taken from the population.

    It is trying to derive the mean and variance, i have to state which is the best and worst estimator out of the 3 given.

    That is all the information in the question im afriad.

    You need to apply the following formulae:

    1. E(aX_1 + bX_2 + \, .... ) = a E(X_1) + b E(X_2) + \, ....

    2. Var(aX_1 + bX_2 + \, .... ) = a^2 Var(X_1) + b^2 Var(X_2) + \, ....

    when X_1, \, X_2, \, .... are independent (which they are here).

    For example:


    E\left( \frac{X_1 + X_2 + \, .... \, + X_n}{n - 1} \right) = \frac{1}{n-1} E(X_1) + \frac{1}{n-1} E(X_2) + \, .... \, + \frac{1}{n-1} E(X_n)

     = \frac{1}{n-1} \left (E(X_1) + E(X_2) + \, .... + E(X_n) \right) = \frac{1}{n-1} (\mu + \mu + \, .... + \mu)

     = \frac{n \mu}{n-1}.


    Var\left( \frac{X_1 + X_2 + \, .... \, + X_n}{n - 1} \right)

     = \frac{1}{(n-1)^2} Var(X_1) + \frac{1}{(n-1)^2}Var(X_2) + \, .... \, + \frac{1}{(n-1)^2} Var(X_n)

     = \frac{1}{(n-1)^2} \left (Var(X_1) + Var(X_2) + \, .... + Var(X_n) \right) = \frac{1}{(n-1)^2} (\sigma^2 + \sigma^2 + \, .... + \sigma^2)

     = \frac{n \sigma^2 }{(n-1)^2}.


    Quote Originally Posted by bill2010 View Post
    [snip]i have to state which is the best and worst estimator out of the 3 given[snip]
    You don't say "best and worst" in which sense ....
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