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Suppose that f(x) = P aixi and g(x) = P bjxj are primitive in Rx Suppose p is irreducible in R There is a first index r such that p does not divide ar and a first index s such that p does not divide bs The coefficient of xrs in f(x)g(x) is arbs P P iB o liv ia D re x l S p ed B o e F a rn e r L cu s t L y m an D u f o r d Le l an d R d a Hu b e rt F a is o n J a p e r T o p to n G ri fto n A u ro r B is co e Tu rk e y C li m a x To p s ai l R wa n E n fie l d S il o a m M ria h H l f x B a ke rs El e rb e S a lu d a W il m a r Ga rn er V n n BF(x) = P(X = x) satis es the following conditions 1 f(x) 0 2 P f(x) = 1 Example A box contains 5 balls numbered 1;2;3;4;

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1, if x ≤ a A simple computation yields E(X) = ab 2 Var(X) = (b−a)2 12 M(s) = esb −esa s(b−a) When a = 0 and b = 1, this is known as the uniform distribution over the unit interval, f(x) = 1 1F(xjB) = P(X = xjB) = P(X = x;B) P(B) and the conditional expectation of X given B is EXjB = X x xf(xjB) 1 The partition theorem says that if Bn is a partition of the sample space then EX = X n EXjBnP(Bn) Now suppose that X and Y are discrete RV's If y is in the range of Y then Y = y is# * ' * # ) $ # * /' # * 0 1 !

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