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20121ee132A_1_EE132A_Problems_HW1

# 20121ee132A_1_EE132A_Problems_HW1 - Communication Systems...

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Unformatted text preview: Communication Systems EE 132A UCLA Winter quarter 2011/2012 Prof. Suhas Diggavi Handout # 4, Due: Thursday, January 19, 2012 Homework Set # 1 Problem 1 (Basic Probabilities) Find the following probabilities. (a) A box contains m white and n black balls. Suppose k balls are drawn. Find the probability of drawing at least one white ball. (b) We have two coins; the first is fair and the second is two-headed. We pick one of the coins at random, we toss it twice and heads shows both times. Find the probability that the coin is fair. Hint: Define X as the random variable that takes value 0 when the coin is fair and 1 otherwise. Problem 2 (Correlation versus Independence) Let Z be a random variable with p.d.f.: f Z ( z ) = 1 / 2 ,- 1 ≤ z ≤ 1 , otherwise . Also, let X = Z and Y = Z 2 . (a) Show that X and Y are uncorrelated. (b) Are X and Y independent? (c) Now let X and Y be jointly Gaussian, zero mean, uncorrelated with variances σ 2 X and σ 2 Y respectively. Are X and Y independent? Justify your answer.independent?...
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20121ee132A_1_EE132A_Problems_HW1 - Communication Systems...

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