Let now a c a then because cos is monotone

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Unformatted text preview: − λt)e−λt , dλ the likelihood is increasing in λ for 0 ≤ λ ≤ 1 , and it is decreasing in λ for λ ≥ 1 , so the likelihood t t is maximized at 1 . That is, λM L (t) = 1 . t t Example 3.7.2 Suppose it is assumed that X is drawn at random from the uniform distribution on the interval [0, b], where b is a parameter to be estimated. Find the ML estimator of b given X = u is observed. (This is the continuous-type distribution version of Example 2.8.2.) Solution: The pdf can be written as fb (u) = 1 I{0≤u≤b} . Recall that I{0≤u≤b} is the indicator b function of {0 ≤ u ≤ b}, equal to one on that set and equal to zero elsewhere. The whole idea now is to think of fb (u) not as a function of u (because u is the given observation), but rather, as a 1 function of b. It is zero if b < u; it jumps up to u at b = u; as b increases beyond u the function decreases. It is thus maximized at b = u. That is, bM L (u) = u. 3.8 3.8.1 Functions of a random variable The distribution of a function of a random variable Often one random variable is a func...
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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 Tech.

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