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Suppose $X_1,\ldots,X_n$ are a random sample of distribution with probability density function

$$ f(x) = \begin {cases} \frac{\theta}{2} & \text{}\ x=-1,1 \\ 1-\theta & \text{} \ x=0 \end {cases},0<\theta<1 $$

Determine the MLE of parameter $\theta$

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closed as too localized by Did, William, Noah Snyder, Norbert, Nate Eldredge Oct 7 '12 at 17:22

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What did you try? Where are you stuck? – Davide Giraudo Sep 11 '12 at 11:45
You asked 46 questions, I believe you added motivation, what you tried, where you were stuck, to none of them. On second thought, I prefer to delete my answer. – Did Sep 11 '12 at 16:35
up vote 0 down vote accepted

First of all this is a discrete probability distribution and is not a density function.You need to know the observed values for $X$$_i$ for i=1,2,...,n. Then for the likelihood use terms in the product that are θ/2 when $X$$_i$=-1 or $X$$_i$=1 and 1-θ when $X$$_i$=0. You could look at this like a binomial where θ is the probability of a 1 or -1 and and 1-θ is the probability of a 0. Then the mle for θ would be the proportion of times a 1 or a -1 comes up.

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It is the proportion of zeros in your sample.

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Actually 1-proprtion of zeros in the sample. – Michael Chernick Sep 11 '12 at 14:48

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