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ps4-questions

# ps4-questions - Middle East Technical University Faculty of...

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Middle East Technical University Faculty of Economics and Administrative Sciences Department of Economics Econ 206 – Spring 2008 PROBLEM SET # 4 1. Let X be a Bernoulli random variable. The probability function is ( ) ( ) 1 1 , 0,1 0 , x x p p x p x otherwise - - = = Find MLE of the parameter p. 2. Let X 1, …,.X n be random sample with pdf f(x; θ )= θ 2 (1+x)(1- θ ) x where E(X)= θ θ ) 1 ( 2 - and Var(X)= 2 ) 1 ( 2 θ θ - for 0< θ <1, x>0. a. Find MLE of θ . b. Find MLE of E(X) and V(X). 3. Suppose that t is a random variable with a distribution ( ) , t 0 t f t e λ λ - = . Find the MLE of λ . 4. Suppose 1 16 ,..., X X is a random sample of size n=16 from a normal distribution, i X ~ N( ,1), μ and we wish to test 0 : 20 H μ = at significance level 0.05, α = based on the sample mean X . a. Find the probability of a Type II error for the alternative : 21 a H μ = . b. Find the probability of a Type II error for the alternative : 19 a H μ = . 5. A single observation of a random variable having an exponential distribution is used to test the null hypothesis is that the mean of the distribution is 2 θ = against that it is 5 θ = . If the null hypothesis is accepted if and only if the observed value of the random variable is less

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ps4-questions - Middle East Technical University Faculty of...

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