Unformatted text preview: X,Z ). (c) Find Var[ Z  Y ]. Problem 4. Suppose X has pdf f X ( x ) = ( θx θ1 if 0 < x < 1 , otherwise . Here, θ > 0. (a) Find the cdf F X of X . (b) Find EX . (c) Find the likelihood function f x 1 ,...,x n  θ for this distribution given a random sample x 1 ,...,x n . (d) Find the maximum likelihood estimator ˆ θ MLE of θ . Problem 5. Suppose X 1 ,...,X n are iid and have a distribution depending upon a parameter θ ∈ R . Suppose ˆ θ n is an estimator for θ (based on a sample of size n ) whose pdf is given by f ˆ θ ( t ) = √ n 2 e√ n  tθ  . Suppose 0 < α < 1. Find a such that [ ˆ θa, ˆ θ + a ] is a (1α )level conﬁdence interval for θ ....
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 Fall '07
 EHRLICHMAN
 Normal Distribution, Probability theory, probability density function, Maximum likelihood, Estimation theory, Likelihood function

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