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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
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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