By Van Der Merwe A. J., Du Plessis J. L.

**Read Online or Download A Bayesian Approach to Selection and Ranking Procedures: The Unequal Variance Case PDF**

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**Extra resources for A Bayesian Approach to Selection and Ranking Procedures: The Unequal Variance Case**

**Example text**

Such an estimate is called the maximum-likelihood estimate, which will be discussed in Chapter 18. Since the numbers are very large, we investigate the following ratio: K NÀK , K NÀ1ÀK k nÀk k nÀk Pk {N} ¼ N NÀ1 Pk {N À 1} n n N ÀK NÀ1 nÀk n NÀn NÀK ¼ ¼ N NÀ1ÀK N NÀKÀnþk n nÀk 48 Discrete Distributions The ratio (Pk {N})=(Pk {N À 1}) is greater than 1 if (N 2 n)(N 2 K ) . N (N 2 K 2 n þ k), Nk , nK, or N , nK/k. Conclusion. If N is less than nK/k, then Pk(N) is increasing, and if N is greater than nK/k, then Pk(N ) is decreasing.

Thus we can write the following conditional probabilities for the host opening door B: 1 P{B j A} ¼ ; P{B j B} ¼ 0; P{B j C} ¼ 1 2 We can now calculate the total probability P(B) of the host opening door B. P{B} ¼ P{B j A}P{A} þ P{B j B}P{B} þ P{B j C}P{C} ¼ 1 1 1 1 1 Á þ0Á þ1Á ¼ 2 3 3 3 2 Using Bayes’ theorem [Eq. 4)] we can now find the a posteriori probabilities of the car behind the doors A or C (B has already been opened by the host) conditioned on B: 1 1 P{BA} P{B j A}P{A} 2 Á 3 1 P{A j B} ¼ ¼ ¼ ¼ 1 P{B} P{B} 3 2 1 P{BC} P{B j C}P{C} 1 Á 3 2 P{C j B} ¼ ¼ ¼ ¼ 1 P{B} P{B} 3 2 Similar analysis holds for other cases of opening the doors A or C.

Since getting k red ballsÀ and Á 40 (10 À 602Ák) blue balls are functionally independent, the total number of ways are k Á 10 À k using the product rule. 3 MULTINOMIAL DISTRIBUTION The binomial distribution of the previous section can be generalized to n repeated independent trials where each trial has m outcomes fEi, i ¼ 1, . . , mg with probabilities PfEig ¼ fpi, i ¼ 1, . . , mg and p1 þ p2 þ Á Á Á þ pm ¼ 1 (4:3:1) 42 Discrete Distributions Each of the outcomes fEi, i ¼ 1, . . , mg occurs fki, i ¼ 1, .