problem_set4

# problem_set4 - EE 511 Problem Set 4 1. Consider a Gaussian...

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EE 511 Problem Set 4 1. Consider a Gaussian random variable X with E [ X ] = 0 and var ( X ) = 1. (i) Determine the moment generating function φ X ( s ), and (ii) Using the result of part (i), determine the ChernoF bound for P [ X a ]. (iii) Use the Chebyshev inequality P [ | X - E [ X ] | ≥ δ ] var ( X ) 2 , to derive another bound for P [ X a ]. 2. Consider two zero-mean random variables X and Y . Let Z = X + aY . Suppose that Z is independent of Y . Show that E [ X | Y = y ] = - ay . 3. Let X be a uniform random variable in [0 , 100]. Determine E [ X ] and E [ X | X 65]. 4. Let X be a Poisson random variable with probability mass function P X ( k ) = e - a a k k ! for k = 0 , 1 , 2 , ··· . Determine E [ X ] and V ar ( X ). 5. Consider the random variable X with pdf f X ( x ) = ( λe - λx x 0 0 else Determine E[X], f X ( x | X 2), and E [ X | X 2]. 6. Let Y be a random variable. (i) We want to choose a constant c as an estimate of Y such that E [( Y - c ) 2 ] (mean squared error) is minimized.

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## This note was uploaded on 10/26/2011 for the course ECE 171 taught by Professor Tu during the Spring '11 term at Aarhus Universitet.

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problem_set4 - EE 511 Problem Set 4 1. Consider a Gaussian...

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