Finding the MLE of a Poisson Distribution

by laura_a
Tags: distribution, poisson
 P: 61 1. The problem statement, all variables and given/known data Suppose that X has a poisson distribution with parameter $$\lambda$$. Given a random sample of n observations, find the MLE of $$\lambda$$, $$\hat{\lambda}$$. 2. Relevant equations The MLE can be found by $$\Sigma^{n}_{i=1} \frac{e^{- \lambda} \lambda^{x_{i}}}{x_{i}!}$$ = $$e^{- \lambda} \times ( \frac{\lambda^{x_1}}{x_{1}!} + \frac{\lambda^{x_2}}{x_{2}!} + . . . )$$ =e^{- \lambda} \Sigma \lambda^{x_i} \div x_{i}! (couldn't get the latex to work from this line) =e^{- \lambda} \Sigma \lambda^{x_i} \times \frac{1}{x_{i}!} Now this is the bit that I think i've stuffed up - if not already = - \lambda log(y) \times x_i \times log(\frac{1}{x_{i}} I know what the answer is meant to be, I was given it in the text \hat{\lambda} = \bar{x} Can anyone tell me where I went wrong, or if I'm even close to being on the right track. I'm a correspondance student so pretty much learning it from reading the notes and the text book which has no examples! Thanks heaps 3. The attempt at a solution

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