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HW2d - Advanced Digital Signal Processing Homework 2 1 Let...

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Unformatted text preview: Advanced Digital Signal Processing Homework 2 1. Let { } n X X , , 1 K be an i.i.d. sample with marginal p.d.f. ( ) , ; > = − x x e x f θ θ θ . a. Show that the maximum likelihood estimator (MLE) for θ is X 1 , where X is the sample mean. b. Show that the CR bound ( ) θ 1 − I for θ given { } n X X , , 1 K is of the form: ( ) n I 2 1 θ θ = − . c. Now, using the fact that for large n the MLE is distributed as approximately ( ) ( ) θ θ 1 , − I N , show that − − − + n Z X n Z X 2 1 1 1 , 2 1 1 1 α α is a ( ) % 100 . 1 α − confidence interval for θ , where ( ) p Z is the p-th quantile of ( ) 1 , N . 2. For the same exponential model for { } n X X , , 1 K as in problem (1): a. Show that the Neyman Pearson most powerful (MP) test of level α for the simple hypotheses : θ θ = H and 1 1 1 , : θ θ θ θ > = H is of the form: Give an equation for the threshold γ of the test in terms of...
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