# Isye 2027

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Unformatted text preview: or each value of θ, then recall from the interpretation, (3.2), of a pdf, that for suﬃciently small, fθ (u) ≈ 1 P u − 2 < X < u + 2 . That is, fθ (u) is proportional to the probability that the observation is in an -width interval centered at u, where the constant of proportionality, namely 1 , is the same for all θ. Following tradition, in this context, we call fθ (u) the likelihood of the observation u. The maximum likelihood estimate of θ for observation u, denoted by θM L (u), is deﬁned to be the value of θ that maximizes the likelihood, fθ (u), with respect to θ. 98 CHAPTER 3. CONTINUOUS-TYPE RANDOM VARIABLES Example 3.7.1 Suppose a random variable T has the exponential distribution with parameter λ, and suppose it is observed that T = t, for some ﬁxed value of t. Find the ML estimate, λM L (t), of λ, based on the observation T = t. Solution: The estimate, λM L (t), is the value of λ > 0 that maximizes λe−λt with respect to λ, for t ﬁxed. Since d(λe−λt ) = (1...
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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.

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