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  1. #1
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    Probability & Statistics

    A random sample of two functions x1, x2, is drawn from the distribution with pdf:

    2/θ^2 (θ - x) 0<x<θ
    f(x;θ) = {
    0 otherwise

    find the method of moments estimator of θ and show that it is not bias.


    Please help!
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  2. #2
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    Quote Originally Posted by James1220 View Post
    A random sample of two functions x1, x2, is drawn from the distribution with pdf:

    2/θ^2 (θ - x) 0<x<θ Mr F says: I assume you mean {\color{red} \frac{2 (\theta - x}{\theta^2}}.
    f(x;θ) = {
    0 otherwise

    find the method of moments estimator of θ and show that it is not bias.


    Please help!
    Calculate E(X) as a function of \theta using the given pdf.

    Note that the sample mean is \bar{X} = \frac{x_1 + x_2}{2}. Estimate \theta by equating E(X) to the sample mean. I get \hat{\theta} = \frac{3(x_1 + x_2)}{2} = 3 \bar{X}.

    Now show that E(\hat{\theta}) = \theta.
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  3. #3
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    Do i get the mean E(X) by integrating? Im not managing to get it so far. I got 1 - (2x)/theta
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  4. #4
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    You should take this integral:

    E(x)=∫xf(x)dx=∫x[(2/(Q)(Q-x)]dx

    The result is E(x)=Q/3
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  5. #5
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    Quote Originally Posted by James1220 View Post
    Do i get the mean E(X) by integrating? Im not managing to get it so far. I got 1 - (2x)/theta
    E(X) = \int_0^{\theta} x \, \frac{2 (\theta - x)}{\theta^2} \, dx = \frac{2}{\theta^2} \int_0^{\theta} x \, (\theta - x) \, dx.
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  6. #6
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    Thank you Mr F and selinunan. I have finally integrated and found E(X)! And yes it is θ/3. When showing that it is unbiased, theta hat = 3 xbar. How do i go about finding E(theta hat)?
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  7. #7
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    Ooo never mind! I've found it thanks!
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  8. #8
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    Quote Originally Posted by James1220 View Post
    Ooo never mind! I've found it thanks!
    A random sample of two functions x1, x2, is drawn from the distribution with pdf:

    2/θ^2 (θ - x) 0<x<θ
    f(x;θ) = {
    0 otherwise

    find the method of moments estimator of θ and show that it is not bias.

    With help of fellow mathematicians, i found the solution to the above problem. the following question relates to this and is the second part of the question :

    Show that the maximum likelihood estimator of θ is given by:

    \hat{\theta} = 1/4 [3(x_1 + x_2) + Square root of (9x_1^2 - 14x_1x_2 + 9x_2^2)]

    Im troubled because of the fact that there are 2 variables x_1 and x_2, so i'm having trouble finding how to fond the mle. Help if you can Pleaseee!
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  9. #9
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    Quote Originally Posted by James1220 View Post
    A random sample of two functions x1, x2, is drawn from the distribution with pdf:

    2/θ^2 (θ - x) 0<x<θ
    f(x;θ) = {
    0 otherwise

    find the method of moments estimator of θ and show that it is not bias.

    With help of fellow mathematicians, i found the solution to the above problem. the following question relates to this and is the second part of the question :

    Show that the maximum likelihood estimator of θ is given by:

    \hat{\theta} = 1/4 [3(x_1 + x_2) + Square root of (9x_1^2 - 14x_1x_2 + 9x_2^2)]

    Im troubled because of the fact that there are 2 variables x_1 and x_2, so i'm having trouble finding how to fond the mle. Help if you can Pleaseee!
    Read this thread carefully: http://www.mathhelpforum.com/math-he...estimator.html

    Now try the question again.

    Post if you're still stuck (please be sure to say where you're stuck).
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