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Handbook of mathematics for engineers and scienteists part 155. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | 1046 Probability Theory 20.2.2-6. Characteristic functions. Semi-invariants. 1 . The characteristic junction of a random variable X is the expectation of the random variable eitX i.e. f t E eitX f eitx dF x J-CO 2 eitxj Pj in the discrete case eitxp x dx in the continuous case J-CO where t is a real variable ranging from -to to to and i is the imaginary unit i2 -1. Properties of characteristic functions 1. The cumulative distribution function is uniquely determined by the characteristic function. 2. The characteristic function is uniformly continuous on the entire real line. 3. f t f 0 1. 4. f-t f t . 5. f t is a real function if and only if the random variable X is symmetric. 6. The characteristic function of the sum of two independent random variables is equal to the product of their characteristic functions. 7. If a random variable X has a kth absolute moment then the characteristic function of X is k times differentiable and the relation f m 0 imE Xm holds for m k. 8. If x1 and X2 are points of continuity of the cumulative distribution function F x then F x2 - F x1 lim ---------- 2 f t dt. 20.2.2.12 2n T oj__t it 9. If f-O f t dt to then the cumulative distribution function F x has a probability density function p x which is given by the formula p x e itxf t dt. 20.2.2.13 2n 7-00 If the probability distribution has a fcth moment ak then there exist semi-invariants cumulants t1 . Tk determined by the relation it 1 ln f t Tr i o tk . 1 1 20.2.2.14 The semi-invariants t1 . Tk can be calculated by the formulas _ -1 d1 ln f t I T1 1 dt1 L 20.2. Random Variables and Their Characteristics 1047 20.2.2-7. Generating functions. The generating function of a numerical sequence a0 a1 . is defined as the power series tt X z 2 pZ n 0 20.2.2.15 where z is either a formal variable or a complex or real number. If X is a random variable whose absolute moments of any order are finite then the series e X n n n 0 20.2.2.16 is called the moment-generating function of the random .
