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By Howard G. Tucker and Ralph P. Boas (Auth.)

ISBN-10: 1483200116

ISBN-13: 9781483200118

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Extra resources for An Introduction to Probability and Mathematical Statistics

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Assuming the above factorization to be true, we obtain Sec. 1] F ,, X ABSOLUTELY CONTINUOUS DISTRIBUTIONS . . 5. We now prove the condition is necessary. Assuming the random variables to be independent we find that for every (x\, • • •, xn) n By the very definition of a joint density, the integrand of the last multiple integral above is a joint density, and the theorem is proved. Conditional distributions are needed in much work in probability and statistics, and the remainder of this section is devoted to these.

K Xk 42 RANDOM VARIABLES; PROBABILITY DISTRIBUTIONS [Chap. 3 which establishes the condition. We now prove that the equation in the theorem implies independence of the random variables. We first remark that if (ii, i2 , • • •, ik) is a subset of the integers (1, 2, • • • , ra), then k FxiuXit, . . X u f e , XitJ• • •, Xik) = J J FxiiiXij). J =I This easily follows from the condition given in the theorem by taking the limit of both sides as xa—> oo where a {ii, i2 , • • • , ik} and by using The­ orem 2 and Theorem 1 of this section.

Xn(%h * * * ? Xm \xm +i, ' ' ', Xn) By the very definition of conditional probability and by the multiplication rule we have fXx,Xt(x h X) = fxi\Xt(Xi\x )fxt(X2) = \Xi(x2\xi) fxi(xi)fXt 2 and 2 fxx,Xt,xAXi, X2jXZ ) = fxi(Xi)fxt\Xi(x2\Xi)fxt\Xi,Xt(X3\xi, x2). This multiplication rule for conditional densities is now used to obtain the multivariate hypergeometric distribution. Suppose an urn contains r red balls, w white balls, and b blue balls. Let us suppose that n balls are selected without replacement, where 1 ^ n ^ r + w + b.

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An Introduction to Probability and Mathematical Statistics by Howard G. Tucker and Ralph P. Boas (Auth.)

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