chapter 6 - Chapter 6 Limit Theorems In this chapter we...

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Chapter 6 Limit Theorems In this chapter, we consider various bounds on probabilities that don’t depend on specific forms of CDFs/pdfs/pmfs Relative frequencies and large numbers of trials Laws of large numbers Asymptotic distributions of random variables P{ X > u} = 1 – F X (u) 0 as u Given the pdf/pmf/CDF, we can find P{ X > 5} exactly What can be said about P{ X > 5} when we don’t know the probabilistic description? How small is P{ X > 5}? Fundamental idea: If g(x) h(x) for all x (a,b), then a b g(x)dx a b h(x)dx Markov’s Inequality is an upper bound on P{ X > α } for non-negative random variables with finite mean μ Markov’s Inequality: If X is a non-negative random variable with finite mean μ , then, for any α > 0, P{ X > α } μ α The bound is uninteresting for α μ Bound 0 as α Proof: P{ X > α } = a f(u)du = 0 g(u)f(u)du where g(u) = 1, u > α , 0, elsewhere. α α u u g(u) 1 f(u) P{ X > α } = 0 g(u)f(u)du 0 h(u)f(u)du where h(u) = u/ α α u 1 α 1 h(u) = u/ α u P{ X > α } 0 (u/ α )f(u)du = μ α P{ X > α } μ / α © 1997 by Dilip V. Sarwate. All rights reserved. No part of this manuscript may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise without the prior written permission of the author.
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Chapter 6 Limit Theorems 159 Because the bound is so general, it can be applied when very little is known about the distribution Because the bound is so generally applicable, it is often quite loose Example: P{ X > 10} 10 –1 for all random variables with average value 1 If X is an exponential random variable with parameter 1, i.e., μ = 1, then P{ X > 10} = exp(–10) 4.54 × 10 –5 << 10 –1 But, do we know the distribution of X ? For λ > 0, the Chernoff bound uses exp λ (u– α ) as an upper bound on the step function g P{ X > α } 0 exp λ (u– α )f(u)du = exp(– λα ) E[exp( λ X )] For exponential RV, P{ X > α } exp(– λα )/(1– λ ) Example: A bit transmitted over a data link is received incorrectly with probability p.
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