Discrete type random variables decision rule is the

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Unformatted text preview: uires the computation of joint probabilities such as P ({X = 1} ∩ H1 ). For brevity we write this probability as P (H1 , X = 1). Such probabilities cannot be deduced from the likelihood matrix alone. Rather, it is necessary for the system designer to assume some values for P (H0 ) and P (H1 ). Let the assumed value of P (Hi ) be denoted by πi , so π0 = P (H0 ) and π1 = P (H1 ). The probabilities π0 and π1 are called prior probabilities, because they are the probabilities assumed prior to when the observation is made. 2.11. BINARY HYPOTHESIS TESTING WITH DISCRETE-TYPE OBSERVATIONS 57 Together the conditional probabilities listed in the likelihood matrix and the prior probabilities determine the joint probabilities P (Hi , X = k ), because P (Hi , X = k ) = πi pi (k ). The joint probability matrix is the matrix of joint probabilities P (Hi , X = k ). For our first example, suppose π0 = 0.8 and π1 = 0.2. Then the joint probability matrix is given by H1 H0 X=0 X=1 X=2 X=3 0.00 0.02 0.0...
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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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