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Unformatted text preview: ECEN 303: Assignment 11 Problems: 1. Jensen inequality. A twice differentiable real-value function f of a single variable is called convex if its second derivative ( d 2 f/dx 2 )( x ) is nonnegative for all x in its domain of definition. (a) Show that the functions f ( x ) = e x , f ( x ) =- ln x , and f ( x ) = x 4 are all convex. (b) Show that if f is twice differentiable and convex, then the first order Taylor approxima- tion of f is an underestimate of the function, that is, f ( a ) + ( x- a ) df dx ( a ) f ( x ) , for every a and x . (c) Show that if f has the property in part (b), and if X is a random variable, then f (E[ X ]) E[ f ( X )] . 2. In order to estimate f , the true fraction of smokers in a large population, Alvin selects n people at random. His estimator M n is obtained by dividing S n , the number of smokers in his sample, by n , i.e., M n = S n /n . Alvin chooses the sample size n to be the smallest possible number for which the Chebyshev inequality yields a guarantee that...
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- Fall '07