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

ISBN-10: 1483200116

ISBN-13: 9781483200118

**Read or Download An Introduction to Probability and Mathematical Statistics PDF**

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**Example text**

W . (n — Xi — x2 — • • • xk-i\ n — xi — • • • — Xk-i trials lor Ak to occurAis ( J. \ xk / Xk Sec. 5] MULTIVARIATE DISCRETE DENSITIES 53 Thus where xk+i = n — X\ — • • • — Xk. The reader can easily check (by can celing) that Consequently where 0 ^ Xi ^ n, 0 ^ x2 ^ n — xh • • •, and 0 ^ xk ^ n — xx xk-i. The probability distribution given above is called the multinomial distribution. EXERCISES 1. An urn contains r red balls, w white balls, b blue balls, and g green balls. One selects n balls at random without replacement.

This proves the theorem. If X is a random variable and K is any real number, then by KX we mean the function which assigns to every co £ 12 the number KX(co). Formally, KX = {(co, #X(co))|co £12}. 2. Theorem. / / X is a random variable, and if K is any real number, then KX is a random variable. Proof. We consider three cases. (i) K = 0. In this case L 1^12 £ a if J x ^ 0. (ii) K > 0. In this case, for every real x, [KX ix] = [X ^ x/K] £ a. (iii) K < 0. In this case, for every real x, [KX ^ x] = [ X ^ x / K ] £ a.

Then use this fact to deduce problem 2 as a corollary to Theorem 6. 4-10. Prove Theorems 7-13. 4 Distribution Functions Although the notion of random variable is the central notion of probability and statistics, it is not considered of interest by itself. With every random variable is associated another function known as its distribution function. The study of probability (and statistics) thus becomes distinct from a usual course in analysis in that properties of functions (random variables) are considered in terms of corresponding distribution functions.