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MLE of Poisson dist

  1. Nov 19, 2012 #1
    1. The problem statement, all variables and given/known data
    Let [itex] X_1,...,X_n [/itex] be a random sample from a poisson distribution with mean [itex]\lambda[/itex]

    Find the MLE of [itex]\lambda^2 + 1 [/itex]


    2. Relevant equations



    3. The attempt at a solution

    I found [itex]\hat{\lambda}=\bar{x}[/itex]

    Can I just square it and add 1 and solve for lambda hat?

    If not I have no idea how I would get the FOC (with respect to [itex] \lambda^2 + 1 [/itex])

    of the log-likelihood function [itex] \ln{L(\lambda^2+1)}=-n\lambda + \Sigma_{i=1}^n x_i \ln{\lambda} - \ln{\Pi_{i=1}^n x_i!} [/itex]
     
  2. jcsd
  3. Nov 19, 2012 #2

    haruspex

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    That's my understanding of how MLE works. If α is the value of λ that maximises the likelihood of the observed data, then (α2+1) must be the value of λ2+1 that does the same.
     
  4. Nov 19, 2012 #3
    cool. thanks again.
     
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