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

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

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Let X and Y denote the number of red balls and white balls, respectively, in the "sample of size n" We now derive the joint discrete density function using the fact that fx Ax, y) = Clearly fx(x)fY\x(y\x). and Before performing the desired multiplication it might be instructive to enter the details of the determination of the range of X , Y. In the first 52 DISCRETE DISTRIBUTIONS [Chap. 4 place x ^ n. Also x ^ r. Thus x ^ min (n, r). On the other hand, 0 ^ x, and in the formula for fx(x) we see that n — x ^ w + b or x ^ n — w — b.

3. Let Yh F2, • • • , Yn denote n independent random variables, and assume that for each Yi, P[Yi = k] = pk, k — 1, 2, • • • , m + 1, where p x+ p2+ • • • + p m+i = 1. Let Zkdenote the number of Fi's equal to k, k = 1, • • • , m + 1. Find fzy zm (zi, • * • , Zm). 4. What distribution does the multinomial distribution become when k = 1? 5. In the density of the multivariate hypergeometric distribution derived in this section, rewrite the range of X, Y so that the range of Y does not depend on x.

They are easier to handle than the others if one is not familiar with the theory of measure and the Lebesgue integral. " Another unfortunate aspect of this false dichotomy is that the distribu­ tion functions of the so-called "continuous random variables" (whatever this expression literally means) are only a small part of those distributions with continuous distribution functions. ) The main point to be brought out by the last unparenthesized sentence is that the word "continuous" is used in a particular sense here.

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