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# ps3 - ECE 804 Random Signal Analysis OSU Autumn 2009...

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ECE 804, Random Signal Analysis Oct. 12, 2009 OSU, Autumn 2009 Due: Oct. 19, 2009 Problem Set 3 Problem 1 We consider a random selection of coins, where the probability of heads, C for the coins is a random variable whose pdf is f C ( c ) = k · c for 0 c 1 and zero otherwise. (a) Find k . (b) Find P ( H ) for a single coin toss. (c) Let E be the event that we choose a coin, flip it n times, and heads appears k of these times. Find P ( E ). (d) Find the conditional pdf f C | E ( c | E ). (e) Find the probability that heads appears in the ( n + 1)st toss given that k heads has appeared in the first n tosses. Useful fact: R 1 0 x m (1 - x ) n dx = m ! n ! ( m + n +1)! , m, n > 0 Problem 2 Consider a random variable X that is passed through a system that has gain with clipping, defined by y = f ( x ) = g · x, | x | ≤ a ga, x a - ga, x ≤ - a where g and a are given positive constants. (a) Find F Y ( y ) in terms of F X ( x ). Sketch F Y ( y ) for a given generic F X ( x ) (such as a Gaussian CDF). (b) Find f Y ( y ) in terms of f X ( x ). Sketch f Y ( y ) for a given generic f X ( x ) (such as a Gaussian pdf).

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ps3 - ECE 804 Random Signal Analysis OSU Autumn 2009...

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