# HW09 - University of Illinois Fall 2009 ECE 313 Problem Set...

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University of Illinois Fall 2009 ECE 313: Problem Set 9: Solutions Continuous Random Variables 1. [Validity of PDFs] If f ( u ) is a nonnegative (or nonpositive) function with ﬁnite nonzero area A , then A - 1 · f ( u ) is a valid pdf. This does not work if f ( u ) takes on both positive and negative values. (a) f ( u ) = 2 u, 0 < u < 1 is a valid pdf. (b) f ( u ) = | u | , | u | < 1 2 is not a valid pdf, but 4 · f ( u ) is. (c) f ( u ) = 1 - | u | , | u | < 1 is a valid pdf. (d) f ( u ) = ln u, 0 < u < 1 is not a valid pdf but - f ( u ) is. (e) f ( u ) = ln u, 0 < u < 2 is not a valid pdf, nor is C · f ( u ) a valid pdf for any choice of C . (f) f ( u ) = 2 3 ( u - 1) , 0 < u < 3 , is not a valid pdf, nor is C · f ( u ) a valid pdf for any choice of C . (g) f ( u ) = exp( - 2 u ) , u > 0 is not a valid pdf but 2 · f ( u ) is. (h) f ( u ) = 4 e - 2 u - e - u , u > 0 , is not a valid pdf, nor is C · f ( u ) a valid pdf for any choice of C . (i) f ( u ) = exp( -| u | ) , | u | < 1 is not a valid pdf but e/ [2( e - 1)] · f ( u ) is. 2. [Calculating probabilities from pdfs] (a) The pdf is as sketched in the left-hand ﬁgure below. f X ( u ) f X ( u ) uu 11 2 u 1 2 c 1 1 2 4 3 1 3 f X ( u ) Since the area under the curve is obviously c/ 2 and should equal 1, we conclude that c = 2. (b) Since X takes on values only in (0 , 1), P { X > - 0 . 5 } = 1. Note: this question had an unintentional typo in it. We meant to ask for P { X > 0 . 5 } which is the shaded area in the middle ﬁgure above and is obviously 1 4 . (c) P { 6 X 2 > 5 X - 1 } = P { 6 X 2 - 5 X + 1 > 0 } = P { (3 X - 1)(2 X - 1) > 0 } . Now, the product of two quantities is positive only when both are positive or both are negative.

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HW09 - University of Illinois Fall 2009 ECE 313 Problem Set...

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