Here we model the observed data by a discrete type

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Unformatted text preview: ity can be applied to calculate the probability of an event, if there is a partition of the sample space. The law of total probability can also be used to compute the mean of a random variable. The conditional mean of a discrete-type random variable X given an event A is defined the same way as the original unconditional mean, but using the conditional pmf: E [X |A] = ui P (X = ui |A), i and also the conditional version of LOTUS holds: E [g (X )|A] = g (ui )P (X = ui |A). i The law of total probability extends from probabilities to conditional expectations as follows. If E1 , . . . , EJ is a partition of the sample space, and X is a random variable, J E [X |Ej ]P (Ej ). E [X ] = j =1 Example 2.10.4 Let 0 < p < 0.5. Suppose there are two biased coins. The first coin shows heads with probability p and the second coin shows heads with probability q , where q = 1 − p. Consider the following two stage experiment. First, select one of the two coins at random, with each coin being selected with probability one...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.

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