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EMET3007/8012 - Week 7 Tutorial
发布时间:2022-09-27
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EMET3007/8012 - Week 7 Tutorial
Exercise 1. Let X1 , . . . , Xn be iid Bernoulli distributions Ber(p). Find the max- imum likelihood estimator for p.
Exercise 2. Let X1 , . . . , Xn be i.i.d. random samples from the distribution with pdf f (x | θ) where
f (x | θ) = xe−北/θ , x > 0
Find the maximum likelihood estimator for θ .
Exercise 3. Suppose 1000 observations X1 , . . . , X1000 are taken from the N(µ, 1) distribution with unknown µ . Unfortunately, the dataset did not record each observation, but only whether the observation was less than 0. Suppose 400 observations were less than 0, and we wish to find the maximum likelihood estimator for µ .
Hint: To derive the likelihood, define Yi = 1 if Xi < 0 and 0 otherwise. Let Y = Yi . Show that the likelihood function is
L(µ, y) = _1y(00)0、py (1 - p)1000 −y
where p is some function of µ .