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homework9 - r is defined by n n n r S lim ∞ → =...

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C OMER ECE 600 Homework 9 1. Explain why the central limit theorem does not hold if the random variables X k have a Cauchy density. Find the pdf of the random variable ° = = n k k 1 X Y if the X k are iid Cauchy random variables with parameters ° = 0 and ± > 0. 2. The strong law of large numbers states that, with probability 1, the successive arithmetic averages of a sequence of iid random variables converge to their common mean ° . What do the successive geometric averages converge to? That is, find n n k k n 1 1 X lim ± ± ² ³ ´ ´ µ = 3. The amount of time that a certain type of component functions before failing is a random variable with pdf 1 0 2 ) f( < < = x x x Once the component fails it is immediately replaced by another one of the same type. If we let X i denote the lifetime of the i th component to be put in use, then ° = = n i i n 1 X S represents the time of the n th failure. The long-term rate at which failures occur, denoted
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Unformatted text preview: r , is defined by n n n r S lim ∞ → = Assuming that the random variables X i , i ² 1, are independent, determine r . 4. The continuous parameter random process X( t ) = e A t is a family of exponentials depending on the random variable A. Express the mean & ( t ), the autocorrelation function R( t 1 , t 2 ), and the first-order pdf f( x ; t ) of X( t ) in terms of the pdf f A ( a ) of A. 5. The random variable C is uniform in the interval (0, T ), where T is not random. Find the autocorrelation function of X t if X t = u( t – C). 6. A random process has sample functions of the form ) sin( A ) X( t t ω = where A is a random variable uniform on [-1,1], and & is fixed. a. Find E[X( t )]. b. Find the pdf of ). 2 X( π...
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