Probability

A Bayesian predictive approach to determining the number of by Dey D. K., Kuo L., Sahu S. K.

By Dey D. K., Kuo L., Sahu S. K.

This paper describes a Bayesian method of combination modelling and a mode according to predictive distribution to figure out the variety of elements within the combinations. The implementation is finished by utilizing the Gibbs sampler. the tactic is defined in the course of the combinations of standard and gamma distributions. research is gifted in a single simulated and one actual facts instance. The Bayesian effects are then in comparison with the chance method for the 2 examples.

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Either of the following theorems is often useful for this purpose. Theorem 2-5 Let X and Y be continuous random variables and let U ϭ ␾1(X, Y ), V ϭ X (the second choice is arbitrary). Then the density function for U is the marginal density obtained from the joint density of U and V as found in Theorem 2-4. A similar result holds for probability functions of discrete variables. Theorem 2-6 Let f (x, y) be the joint density function of X and Y. Then the density function g(u) of the random variable U ϭ ␾1(X, Y ) is found by differentiating with respect to u the distribution Random Variables and Probability Distributions CHAPTER 2 43 function given by G(u) ϭ P[f1 (X, Y ) Յ u] ϭ 6 f (x, y) dx dy (38) 5 Where 5 is the region for which ␾1(x, y) Յ u.

A) A shelf contains 6 separate compartments. In how many ways can 4 indistinguishable marbles be placed in the compartments? (b) Work the problem if there are n compartments and r marbles. This type of problem arises in physics in connection with Bose-Einstein statistics. 98. (a) A shelf contains 6 separate compartments. In how many ways can 12 indistinguishable marbles be placed in the compartments so that no compartment is empty? (b) Work the problem if there are n compartments and r marbles where r Ͼ n.

15. Prove Theorem 2-1, page 42. The probability function for U is given by g(u) ϭ P(U ϭ u) ϭ P[f(X) ϭ u] ϭ P[X ϭ c(u)] ϭ f [c(u)] In a similar manner Theorem 2-2, page 42, can be proved. 16. Prove Theorem 2-3, page 42. , u increases as x increases (Fig. 2-11). There, as is clear from the figure, we have (1) P(u1 Ͻ U Ͻ u2) ϭ P(x1 Ͻ X Ͻ x2) or (2) u2 x2 1 1 3u g(u) du ϭ 3x f (x) dx Fig. , the slope is positive). 67). The theorem can also be proved if cr(u) Ն 0 or cr(u) Ͻ 0. 17. Prove Theorem 2-4, page 42.

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