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Math Help - Maximum likelihood

  1. #1
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    Maximum likelihood

    An individual taken from a very large biological population is of type A with probability p=0.5(1+t) and of type B with probabiliy 1-p=0.5(1-t).

    a) Suppose that X denotes the number of type A individuals in a random sample of size n. What is the probability that X = x?

    My answer: 0.5^x*(1+t)^x*(1-t)^{(n-x)} Classic binomial situation.

    b)Find the maximum likelihood estimator of t and show that it is unbiased.

    My answer: Find log likelihood, differentiate yields
    2x-n(1+t)/[(1+t)(1-t)]

    setting to zero yields t=(2x-n)/n

    To show unbiased, I need to find E(t). This is an integral including 0.5^x*(1+t)^x*(1-t)^{(n-x)} which I have no idea how to integrate.

    Thanks for your help.
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  2. #2
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    Re: Maximum likelihood

    im not sure your liklihood is correct.

    The likelihood is proportional to 0.5^x(1+t)^x \times 0.5^{n-x}(1-t)^{n-x} = 0.5^n(1+t)^x (1-t)^{n-x}

    this shouldn't affect your final answer as long as you did your algebra correctly (the constant term at the front vanishes anyway).


    To show your final answer is unbiased, evaluate:
    E(\frac{2x-n}{n}) = \frac{2E(x) - n}{n} = \frac{2E(x)}{n} - 1

    You know X is binomial and so E(X) = 0.5n(1+t).
    Im assuming that you are allowed to treat the moments of a binomial distribution as standard results instead of having to derive them every time

    \frac{2 \times 0.5n(1+t)}{n} - 1

    =1+t-1
    =t
    Last edited by SpringFan25; December 19th 2011 at 11:43 AM.
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  3. #3
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    Re: Maximum likelihood

    I forgot I could distribute the expectation. Thanks
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