Analysis of straight-line data - download pdf or read online

By Forman Sinnickson Acton

ISBN-10: 0486617475

ISBN-13: 9780486617473

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X+y(u) = ¢x(u)¢y{u) = exp [>. (eiU - 1)] exp [/-L (e iU - 1)] = exp [(>' + /-L) (eiU - 1)] , which is the characteristic function of the Poisson distribution with parameter >. + /-L. 56 CHAPTER 2. , X 2 , ••. , X k be independent and identically distributed exponential random variables with parameter A. Let Sk = Xl + X 2 + ... + X k be the sum of the random variables X I, X 2 , ••• , X k. The characteristic function of Sk is given by which is the characteristic function of the gamma distribution Sk '" GAM(A, k).

0) and k (k > 0) are parameters, and r(k) is a gamma function of order k: f'V r(k) = 10 00 e-xxk-1dx. 15) Thus, if k is a positive integer, r(k} = (k - I)! 3 shows the density of the gamma distribution X GAM(>", k}, where>.. 5, 1,2, 5. 2 o Fig. 3 2 3 s 4 6 x The densities of X GAM(>", k}, where>.. 5, 1,2,5. 4. 18) respectively. 19) s+A It is finally noted that GAM(A, 1) = EXP(A) which will be interpreted in the following section. , (3), where a (a> 0) and f3 (f3 > 0) are parameters. 21 ) The Wei bull distribution is well-known in reliability theory, where f3 and a are called the shape and scale parameters, respectively.

14) Applying the law of total probability, we have 00 Px(x) = LPxlY(x I y)py(y). 15) y=o Taking the expectation of X in the above, we have E[X] = LXPx(x) x 00 = = 00 L L xPxlY(x I y)py(y) x=Oy=O E[E[X I Y]]. 9) and marginal densities fx(x) and jy(y) are given. 5. MULTIVARIATE DISTRIBUTIONS 51 provided fy (y) > o. The conditional distribution of X given Y = y is defined by ( I) = FxlY x Y J x -00 fxy(a, y) jy(y) da. 19) i: i: i: i: Applying the law of total probability, we have fx(x) = fxlY(x I y)fy(y)dy.

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Analysis of straight-line data by Forman Sinnickson Acton

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