Pxe I. (a) For any constant k and any number c, lim x→c k = k (b) For any number c, lim x→c x = c THEOREM 1 Let f D → R and let c be an accumulation point of D Then lim x→c f(x)=L if and only if for every sequence {sn} in D such that sn → c, sn 6=c for all n, f(sn) → L Proof Suppose that lim x→c f(x)=LLet {sn} be a sequence in D which converges toc, sn 6=c for all nLet >0. (10) Z x a2 x2 dx= 1 2 lnja2 x2j (11) Z x2 a 2 x dx= x atan 1 x a (12) Z x3 a 2 x dx= 1 2 x2 1 2 a2 lnja2 x2j (13) Z 1 ax2 bx c dx= 2 p 4ac b2 tan 1 2ax b p 4ac b2 (14) Z 1 (x a)(x b) dx= 1 b a ln a x b x;.
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C A T I O N // FU NC TI ON AL _C LA SS IF IC AT IO N_ SY ST EM _S TR ET S_ 11 X1 7 MX D S TR E FU NC IOAL L EG ND Ex i s tng I er aF w y Existing Regional Arterial Existing Arterial Existing Major Collector Existing Collector Future Regional Arterial Future A rte ial F u t reMaj oC l c Future Collector Ex ist ng Fr ew ay R mp. 2 Moments and Conditional Expectation Using expectation, we can define the moments and other special functions of a random variable Definition 2 Let X. P(jX EXj ) = P((X EX)2 2) = P(Y 2) EY 2 = var(X) 2 (1) Independence and sum of random variables Two random variables are independent independent if the knowledge of Y does not in uence the results of Xand vice versa This 3.
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Thanks for contributing an answer to Mathematics Stack Exchange!. V g a v o ̃a c x n a p x e c u a a c x l f b @ ̂ Љ. C 07 by Karl Sigman 1 Review of Probability Random variables are denoted by X, Y, Z, etc The cumulative distribution function (cdf) of a random variable X is denoted by F(x) = P(X ≤ x), −∞ < x < ∞, and if the random variable is continuous then its probability density function is denoted by f(x) which is related to F(x) via f(x. 76 42 Probability Generating Functions The probability generating function (PGF) is a useful tool for dealing with discrete random variables taking values 0,1,2,.
