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# HW6 - ECE 302 Homework#6 Due 2/25...

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ECE 302, Homework #6, Due 2/25 http://www.ece.purdue.edu/ chihw/ECE302 09S.html Review of Calclulus: The chain rule. Question 1: Consider the following functions. F ( x ) = Z x 0 f ( s ) ds g ( x ) = F ( h ( x )) = Z h ( x ) 0 f ( s ) ds Express d dx F ( x ) and d dx g ( x ) using f ( x ) and h ( x ). (Hint: You should use the chain rule that d dx g ( x ) = d ds F ( s ) fl fl s = h ( x ) × d dx h ( x ).) Question 2: [Intermediate/Exam Level] (Compare it with HW4Q1). Consider a continu- ous random variable X with the following pdf f X ( x ): f X ( x ) = 1 . 5 e - 3 | x | for all x (1) Consider a discrete “quantizer” Y of the magnitude of X as follows. For any X , if k ≤ | X | < k + 1, then Y = k . For example, if the X value is - π , then Y = 3 since 3 ≤ | - π | < 4. If the X value is 1 . 25, then Y = 1. Find the pmf of the discrete variable Y . Namely, find P ( Y = k ) for k = 0 , 1 , 2 · · · . What type of random variables is Y ? Question 3: [Intermediate/Exam Level] Let X = Y 1 + Y 2 + · · · + Y n be a binomial random variable that results from the summation of n independent Bernoulli random variables Y 1 to Y n . The success probability of Y i is p for i = 1 , · · · , n .

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