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MLE_distributions

# MLE_distributions - Connexions module m13500 1 Maximum...

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Unformatted text preview: Connexions module: m13500 1 Maximum Likelihood Estimation - Examples * Ewa Paszek This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License † Abstract This course is a short series of lectures on Introductory Statistics. Topics covered are listed in the Table of Contents. The notes were prepared by Ewa Paszek and Marek Kimmel. The development of this course has been supported by NSF 0203396 grant. 1 MAXIMUM LIKELIHOOD ESTIMATION - EXAMPLES 1.1 EXPONENTIAL DISTRIBUTION Let X 1 ,X 2 ,...,X n be a random sample from the exponential distribution with p.d.f. f ( x ; θ ) = 1 θ e- x/θ , < x < ∞ ,θ ∈ Ω = { θ ;0 < θ < ∞} . The likelihood function is given by L ( θ ) = L ( θ ; x 1 ,x 2 ,...,x n ) = 1 θ e- x 1 /θ 1 θ e- x 2 /θ · · · 1 θ e- x n /θ = 1 θ n exp- ∑ n i =1 x i θ , < θ < ∞ . The natural logarithm of L ( θ ) is ln L ( θ ) =- ( n )ln( θ )- 1 θ n X i =1 x i , < θ < ∞ . Thus, d [ln L ( θ )] dθ =- n θ + ∑ n i =1 x i θ 2 = 0 . The solution of this equation for θ is θ = 1 n n X i =1 x i = x....
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MLE_distributions - Connexions module m13500 1 Maximum...

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