tailieunhanh - Handbook of mathematics for engineers and scienteists part 156

Handbook of mathematics for engineers and scienteists part 156. 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. | . Random Variables and Their Characteristics 1053 a 2 3X Figure . Probability density a and cumulate distribution b functions of normal distribution for a 1 a 1 4. . Normal distribution. A random variable X has the normal distribution with parameters a a2 see Fig. if its probability density function has the form p x exp x - a 2 2a2 x g -œ œ . The cumulative distribution function see Fig. and the characteristic function have the form F x 1 fx t - a 2 .2- L M- 2ä- dt f t exp iat - a2t21 2 J 1 V2nc and the numerical characteristics are given by the formulas E X a Var X a2 Mode X Med X a 71 0 72 0 f 0 k 2m - 1 m 1 2 . k 2k - 1 a2k k 2m m 1 2 . The linear transformation Y reduces the normal distribution with parameters a a2 and cumulative distribution function F x to the standard normal distribution with parameters 0 1 and cumulative distribution function x - e t2 2 dt v n j- moreover -x 1 - x . Remark 1. The values of the cumulative distribution function x of the standard normal distribution are computed by the function NORMSDIST z in EXCEL software for example for b 2 the function call NORMSDIST 2 returns the value . Remark 2. The values of the cumulative distribution function F x of the normal distribution are computed by the function pnorm x mu sigma in MATHCAD software to compute x one should use pnorm x 0 1 . For example b 2 pnorm 2 0 1 . The probability that a random variable X normally distributed with parameters m a2 lies in the interval a b is given by the formula P a b p a-m Em. Fm r Em - ra-my a a a a a 1054 Probability Theory A normally distributed random variable takes values close to its expectation with large probability this is expressed by the sigma rule P X - m ka 2 1 - k for k 1 for k 2 for k 3. The three-sigma rule is most frequently used. The fundamental role of the normal distribution is due to the fact that under mild assumptions the distribution of a