# A Baire category approach in existence theory of by Pianigiani G.

By Pianigiani G.

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Then X has a hypergeometric distribution with parameters n, M, N. Further, if N is very large, then binomial probabilities can be used to approximate hypergeometric probabilities. 1 (Binomial Approximation). If N is large, then the binomial distribution with parameters n and p = M/N can be used to approximate the hypergeometric distribution with parameters n, M, N. Specifically, Note that if X is the number of special objects in a sequence of n objects chosen with replacement from the urn and if the choice of each sequence is equally likely, then X has a binomial distribution with parameters n and p = M/N.

In this case, In particular, the range is now the positive integers. Example: Negative binomial distribution on 0,1, 2,... Let r be a positive integer and X be the number of failures before the r* success in a sequence of independent Bernoulli experiments with success probability p. Then X is said to be a negative binomial random variable, or to have a negative binomial distribution, with parameters r and p. ) For example, if r = 3 and x = 2, then wnere o represents success ana f represents taiiure.

Xk) and commas are understood to mean the intersection of events. 2. The joint probability density function (joint PDF) (or joint density junction) is obtained from the joint CDF by taking multiple partial derivatives. 3. , Xk are said to be mutually independent (or independent) if or, equivalently, if for all real fc-tuples (X1, X 2 , . . , xk, where F,(JC,-) is the CDF and fi(xi) is the PDF of X/for i = l, 2 , . . , k. , Xk are said to be a random sample from that distribution. 4 Chapter 3.