MIT6_262S11_lec21 - 6.262 Discrete Stochastic Processes L21...

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Random walks Def: Let { X i ; i 1 } be a sequence of IID rv’s, and let S n = X 1 + X 2 + · · · + X n for n 1 . The integer-time stochas­ tic process { S n ; n 1 } is called a random walk, or, specif­ ically, the random walk based on { X i ; i 1 } . Our focus will be on threshold-crossing problems. For example, if X is binary with p X (1) = 1 , p X ( 1) = q = 1 p , then k p Pr { S n k } = if p 1 / 2 . 1 p n =1 2 6.262: Discrete Stochastic Processes 4/27/11 L21: Hypothesis testing and Random Walks Outline: Random walks Detection, decisions, & Hypothesis testing Threshold tests and the error curve Thresholds for random walks and Cherno ff 1
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Detection, decisions, & Hypothesis testing The model here contains a discrete, usually binary, rv H called the hypothesis rv. The sample values of H , say 0 and 1, are called the alternative hypotheses and have marginal probabilities, called a priori probabilities p 0 = Pr { H = 0 } and p 1 = Pr { H = 1 } . Among arbitrarily many other rv’s, there is a sequence Y ( m ) = ( Y 1 , Y 2 , . . . , Y m ) of rv’s called the observation. We usually assume that Y 1 , Y 2 , . . . , are IID conditional on H = 0 and IID conditional on H = 1 . Thus, if the Y n are continuous, m f Y ( m ) ( y ) = f ( y n | ) . H | Y | H n =1 | 3 Assume that, on the basis of observing a sample value y of Y , we must make a decision about H , i.e., choose H = 0 or H = 1 , i.e., detect whether or not H is 1. Decisions in probability theory, as in real life, are not necessarily correct, so we need a criterion for making a choice. We might maximize the probability of choosing correctly, for example, or, given a cost for the wrong choice, might minimize the expected cost. Note that the probability experiment here includes not only the experiment of gathering data (i.e., measuring the sample value y of Y ) but also the sample value of the hypothesis.
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