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Unformatted text preview: (k − λ)λk−1 e−λ ,
so the likelihood is increasing in λ for λ < k and decreasing in λ for λ > k ; the likelihood is
maximized at λ = k. Thus, λM L (k ) = k. If k = 0, the likelihood is e−λ , which is maximized by
λ = 0, so λM L (k ) = k for all k ≥ 0. 2.8 Maximum likelihood parameter estimation Sometimes when we devise a probability model for some situation we have a reason to use a
particular type of probability distribution, but there may be a parameter that has to be selected.
A common approach is to collect some data and then estimate the parameter using the observed
data. For example, suppose the parameter is θ, and suppose that an observation is given with pmf
pθ that depends on θ. If the observed value is k, then the likelihood (i.e. probability) that X = k
is pθ (k ). The maximum likelihood estimate of θ for observation k , denoted by θM L (k ), is the value
of θ that maximizes the likelihood, pθ (k ).
Example 2.8.1 Suppose a bent coin is given to a student. The coin is badly bent, but the student
can still ﬂip the coin and see whether it shows heads or tai...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.
- Spring '08
- The Land