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Math Help - Maximum likelihood function

  1. #1
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    Maximum likelihood function

    I'm doing it for a binomial random variable X with one observation X=x. All I want to know is whether I should write my estimator in terms of X or x.

    I believe it to be X. If that is so, should my working be in terms of x then at the final stage change x to X.

    Thanks
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  2. #2
    Super Member ILikeSerena's Avatar
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    Re: Maximum likelihood function

    Quote Originally Posted by Duke View Post
    I'm doing it for a binomial random variable X with one observation X=x. All I want to know is whether I should write my estimator in terms of X or x.

    I believe it to be X. If that is so, should my working be in terms of x then at the final stage change x to X.

    Thanks
    Hi Duke!

    An estimator is a function that estimates a parameter of your model, which in this case is a binomial distribution that has parameters n and p.
    Typically you would use a sample to do the estimation.

    What you would have is:

    X the random variable
    x a specific (possible) observation
    \bar x the sample mean
    \hat p = {\bar x \over n} the estimator for the parameter p of the binomial distribution.
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  3. #3
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    Re: Maximum likelihood function

    Ok but I mean in general would I want the estimator in terms of the random variable or the specific observations. For example you say (correctly) that p= {\bar x \over n}. But if I wanted E(p), I would do E({\bar X \over n}). So should it really be \hat p =({\bar X \over n})?
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  4. #4
    Super Member ILikeSerena's Avatar
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    Re: Maximum likelihood function

    The notation \bar X represents the mean of n random variables, but I do not think that is what you intend.


    Typically x would represent the sample, and you would write \hat p(x) = {\bar x \over n}.

    Your expectation would be written as: E(\hat p) = E[\hat p(X)] = E[X / n], which is an expectation based on a random variable, of which samples are taken to estimate \hat p.
    Btw, this expection is equal to the parameter p.


    In other words, the choice for the symbol depends on what you're talking about.
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