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Unformatted text preview: Information Theory and Coding-HW 3 V Balakrishnan Department of ECE Johns Hopkins University October 8, 2006 1 Markov inequality and Chebyshevs inequality 1.1 Markov Inequality Let f x ( x ) be the probability distribution function of X. P r ( X ) = Z f x ( x ) dx Z x f x ( x ) dx Z + - x f x ( x ) dx = E ( X ) The first step follows from the definition of probability The second step follow from the fact that x/ 1 for x The third step follows from the fact that f x ( x ) 0 for all x as it is a probability density function. 1.2 Chebyshevs inequality We see that E ( X ) = E [( Y- ) 2 ] = 2 and we are required to compute P r ( | Y- | > ) = P r ( X > 2 ) So we have P r ( | X | > 2 ) P r ( | X | 2 ) E ( | X | ) 2 = 2 2 The second step follows directly from Markov inequality and from the fact that | X | = X as X is defined as ( Y- ) 2 1 1.3 The weak law of large numbers Let us compute the variance of Z n . The mean of it is clearly . We have it to be E h ( Z n- ) 2 i = 1 n 2 n X i =1 E...
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This note was uploaded on 04/05/2011 for the course EE 5368 taught by Professor Staff during the Spring '08 term at UT Arlington.
- Spring '08