20121ee132A_1_EE132A_Solutions_HW1

# 20121ee132A_1_EE132A_Solutions_HW1 - Communication Systems...

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Communication Systems EE 132A UCLA Winter quarter 2011/2012 Prof. Suhas Diggavi Handout # 5, Thursday, January 19, 2012 Solutions: Homework Set # 1 Problem 1 (Basic Probabilities) a) First, we ﬁnd the probability of the complement of the event, namely the probability of drawing only black balls. This probability is equal to P [All k balls are black] = ( n k ) ( m + n k ) . Therefore the probability of drawing at least one white ball is equal to P [At least one ball is white] = 1 - ( n k ) ( m + n k ) . b) Deﬁne the following random variables X = ± 0 If the chosen coin is fair , 1 otherwise , and Y = 00 If both outcomes are tail , 01 If the ﬁrst one is tail, the second one is head , 10 If the ﬁrst one is head, the second one is tail , 11 If both outcomes are head . So having these two random variables deﬁned, we want to compute P [ X = 0 | Y = 11]. So we can write P [ X = 0 | Y = 11] = P [ X = 0 ,Y = 11] P [ Y = 11] = 1 / 2 × 1 / 4 P [ Y = 11] = 1 / 8 P [ Y = 11] . Then for P [ Y = 11] we have P [ Y = 11] = P [ X = 0] · P [ Y = 11 | X = 0] + P [ X = 1] · P [ Y = 11 | X = 1] = 1 / 2 × 1 / 4 + 1 / 2 × 1 = 5 / 8 . So, ﬁnally we have P [ X = 0 | Y = 11] = 1 / 8 5 / 8 = 1 / 5 . 1

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Problem 2 Correlation and Independence (i) We have, ¯ X = E [ X ] = Z -∞ xf X ( x ) dx = Z 1 - 1 zf Z ( z ) dz = Z 1 - 1 z. 1
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## This note was uploaded on 04/03/2012 for the course EE 199 taught by Professor Liu during the Spring '10 term at UCLA.

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20121ee132A_1_EE132A_Solutions_HW1 - Communication Systems...

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