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Math Help - estimation of parameter

  1. #1
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    estimation of parameter

    consider the probability mass function
    p_{X} (x; \theta) = \theta ^{x} (1 - \theta )^{1-x}, x = 0,1
    and suppose we observe a sample of 5 outcomes: 0, 0, 1, 0, 1.
    (a) show how maximum likelihood estimation can be used to estimate the parameter \theta
    (b) show how the method of moments can be used to estimate the parameter \theta
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  2. #2
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    Quote Originally Posted by wik_chick88 View Post
    consider the probability mass function
    p_{X} (x; \theta) = \theta ^{x} (1 - \theta )^{1-x}, x = 0,1
    and suppose we observe a sample of 5 outcomes: 0, 0, 1, 0, 1.
    (a) show how maximum likelihood estimation can be used to estimate the parameter \theta
    (b) show how the method of moments can be used to estimate the parameter \theta
    (a) Well what is the likelihood of exactly 2 1'1 in 5 trials?

    (b) Set the expected number of 1's in 5 trials to the observed result.

    CB
    Last edited by CaptainBlack; December 15th 2010 at 11:18 AM.
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    im still confused on how to start this question?
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  4. #4
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    Quote Originally Posted by CaptainBlack View Post
    (a) Well what is the likelihood of exactly 2 1'1 in 5 trials?
    Consider the 1's as successes then and $$N the number of successes in 5 trials:

    L(\theta|N=2)=k \theta^N (1-\theta)^{5-N}=k \theta^2 (1-\theta)^{3}

    Then the maximum liklyhood estimator is the largest $$\Theta which solves:

    \dfrac{d}{d\theta}L(\theta|N=2)=0

    CB
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  5. #5
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    i got \theta = \frac{2}{5} is this correct?
    also, im still confused as what to do for (b), how do i calculate the expected number of 1's in 6 trials?
    thanks so much!
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  6. #6
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    Quote Originally Posted by wik_chick88 View Post
    i got \theta = \frac{2}{5} is this correct?
    Yes
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  7. #7
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    Quote Originally Posted by wik_chick88 View Post
    iim still confused as what to do for (b), how do i calculate the expected number of 1's in 6 trials?
    thanks so much!
    We are setting p_X(1;\theta)=\theta=2/5

    (we are setting the mean of X to the observed proportion in our sample, which is matching first monents of the distribution of number of 1's in a sample with the observed)

    CB
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  8. #8
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    im confused! how do i know (or guess) whether the 6th outcome will be a 0 or a 1?
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  9. #9
    Grand Panjandrum
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    Quote Originally Posted by wik_chick88 View Post
    im confused! how do i know (or guess) whether the 6th outcome will be a 0 or a 1?
    The 6 is a typo for 5

    CB
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