# HW3s - ECE 534 Elements of Information Theory Fall 2010...

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Unformatted text preview: ECE 534: Elements of Information Theory, Fall 2010 Homework 3 Solutions Problem 3.1 (Johnson Jonaris Gad Elkarim + Yasaman Keshtkarjahromi a) Let X have a probability distribution function f(x) E ( X ) = integraldisplay ∞ xf ( x ) dx = integraldisplay t xf ( x ) dx + integraldisplay ∞ t xf ( x ) dx ≥ integraldisplay ∞ t xf ( x ) dx ≥ t integraldisplay ∞ t f ( x ) dx = t.Pr { X ≥ t } Pr { X ≥ t } ≤ E ( X ) t An example of a random variable which achieves the inequality with equality: X = braceleftbigg t with probability p with probability 1- p b) Y is a R.V. with mean μ and variance σ 2 Pr {| Y- μ | > ǫ } = Pr braceleftbig ( Y- μ ) 2 > ǫ 2 bracerightbig ≤ Pr braceleftbig ( Y- μ ) 2 ≥ ǫ 2 bracerightbig ≤ E ( Y- μ ) 2 ǫ 2 = σ 2 ǫ 2 c) Z 1 ,Z 2 ,...Z n are an i.i.d. R.V. with mean μ and variance σ 2 S n = ¯ Z n = 1 n n summationdisplay i =1 Z i Pr braceleftbig | ¯ Z n- μ | > ǫ bracerightbig = Pr braceleftBigg parenleftBig 1 n n summationdisplay i =1...
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