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MLE of Bernoulli trials

  1. Feb 24, 2012 #1
    1. The problem statement, all variables and given/known data
    Two independent bernoulli trials resulted in one failure and one success. What is the MLE of the probability of success θ is it is know that θ is at most 1/4


    2. Relevant equations
    f(x,θ) = θx (1-θ)1-x


    3. The attempt at a solution
    Now, I know how to find the likelihood and use it to solve for the MLE. But I am not sure how the "θ is at most 1/4" would factor into the equation.

    For a Bernoulli trial: f(x,θ) = θx (1-θ)1-x
    L(θ) = θ(1-θ)
    L'(θ) = 1-2θ ----> equate to zero
    θ(hat) = 1/2 (which is the MLE)

    But what do I do about the fact that "θ is at most 1/4"? Please help
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 25, 2012 #2

    Office_Shredder

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    You're trying to maximize L(θ) on the interval [0,1/4]. Recall a basic calculus fact that the maximum of a function on a closed interval is either a critical point (where the derivative is zero) or an end point
     
  4. Feb 25, 2012 #3
    Thank you.
    In this case, since the function is simply: θ(hat) = 1/2 (a constant)
    The maximum will lie at 1/4
    Correct?
     
  5. Feb 25, 2012 #4

    Office_Shredder

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    The critical point is at 1/2. But the function you want to maximize is L(θ) = θ(1-θ)
     
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