Discrete-time stochastic processes

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Unformatted text preview: tomer waits in the queue. Let Un = Yn−1 − Xn for n ≥ 1. Then for any α > 0, and n ≥ 1, Wn is given by (7.7). Also, Pr {Wn ≥ α} is equal to the probability that the random walk based on {Ui ; i ≥ 1} crosses a threshold at α by the nth trial. Final ly, Pr {W ≥ α} = limn→1 Pr {Wn ≥ α} is equal to the probability that the random walk based on {Ui ; i ≥ 1} ever crosses a threshold at α. Note that the theorem speciﬁes the distribution function of Wn for each n, but says nothing about the joint distribution of successive waiting times. These are not the same as the distribution of successive terms in a random walk because of the reversal of terms above. We shall ﬁnd a relatively simple solution to the probability that a random walk crosses a positive threshold in Section 7.4. From Theorem 7.1, this also solves for the distribution of queueing delay for the G/G/1 queue (and thus also for the M/G/1 and M/M/1 queues). 7.3 Detection, Decisions, and Hypothesis testing Consider a situation in which we make n noisy observations of the outcome of a single binary random variable H and then guess, on the basis of the observations alone, which binary outcome occurred. In communication technology, this is called a detection problem. It models, for example, the situation in which a single binary digit is transmitted over some time interval but a noisy vector depending on that binary digit is received. It similarly models the problem of detecting whether or not a target is present in a radar observation. In control theory, such situations are usually referred to as decision problems, whereas in statistics, they are referred to as hypothesis testing. 2 More precisely, the sequence of waiting times W1 , W2 . . . , have distribution functions FWn that converge to FW , the generic distribution of the given threshold crossing problem with unlimited trials. As n increases, the distribution of Wn approaches FW , and we refer to W as the waiting time in steady state. 7.3. DETECTION, DECISIONS, AND HYPOTHESIS TESTING 285 Speciﬁcally, let H0 and H1 be the names for the two possible values of the binary random variable H and let p0 = Pr {H0 } and p1 = 1 − p0 = Pr {H1 }. Thus p0 and p1 are the a priori probabilities3 for the random variable H . Let Y1 , Y2 , . . . , Yn be the n observations. We assume that, conditional on H0 , the observations Y1 , . . . Yn are IID random variables. Suppose, to be speciﬁc, that these variables have a density f (y | H0 ). Conditional on H0 , the joint density of a sample n-tuple y = (y1 , y2 , . . . , yn ) of observations is given by f (y | H0 ) = n Y i=1 f (yi | H0 ). (7.10) Similarly, conditional on H1 , we assume that Y1 , . . . Yn are IID random variables with a conditional joint density given by (7.10) with H1 in place of H0 . In summary then, the model is that H is a rv with PMF {p0 , p1 }. Conditional on H , Y = (Y1 , . . . , Yn ) is an n-tuple of IID rv’s. Given a particular sample of n observations y = y1 , y2 , . . . , yn , we can evaluate Pr {H1 | y } as Q p1 n f (yi | H1 ) i=1 Q Pr {H1 | y } = Qn . (7.11) p1 i=1 f (yi | H1 ) + p0 n f (yi | H0 ) i=1 We can evaluate Pr {H0 | y } in the same way, and the ratio of these quantities is given by Q p1 n f (yi | H1 ) Pr {H1 | y } = Qi=1 . (7.12) Pr {H0 | y } p0 n f (yi | H0 ) i=1 If we observe y and choose H0 , then Pr {H1 | y } is the resulting probability of error, and conversely if we choose H1 , then Pr {H0 | y } is the resulting probability of error. Thus the probability of error is minimized, for a given y , by evaluating the above ratio and choosing H1 if the ratio is greater than 1 and choosing H0 otherwise. If the ratio is equal to 1, the error probability is the same whether H0 or H1 is chosen. The above rule for choosing H0 or H1 is called the Maximum a posteriori probability detection rule, usually abbreviated as the MAP rule. The rule has a more attractive form (and also brings us back to random walks) if we take the logarithm of each side of (7.12), getting n Pr {H1 | y } p1 X ln = ln + zi ; Pr {H0 | y } p0 i=1 where zi = ln f (yi | H1 ) . f (yi | H0 ) (7.13) The quantity zi in (7.13) is called a log likelihood ratio. Note that zi is a function only of yi , and that this same function is used for each i. For simplicity, we assume that this 3 Statisticians have argued since the beginning of statistics about the validity of choosing a priori probabilities for an hypothesis to be tested. Bayesian statisticians are comfortable with this practice and non-Bayesians are not. Both are comfortable with choosing a probability model for the observations conditional on each hypothesis. We take a Bayesian approach here, partly to take advantage of the power of a complete probability model, and partly because non-Bayesian results, i.e., results that do not depend on the a priori probabilies are much easier to derive and interpret within a full probability model. As will be seen, the Bayesian approach also makes it natural to incorporate the results of early observations into updated a priori probabilities for analyzing later observations. 286 CHAPTER 7. RANDOM WALKS, LARGE DEVIATIONS, AND MARTINGALES function is ﬁnite...
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.

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