某快递公司信息化规划方案.ppt
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1、 density that satisfi es this usual integral relation, it must be a density with respect to (k1)-dimensional Lebesgue measure. Hence, technically, the density should be a function of k 1 of the k variables, with the k-th variable implicitly equal to one minus the sum of the others, so that all k var
2、iables sum to one. The choice of which k 1 variables to use in the density is arbitrary. For example, one way to write the density is as follows: f(q1,q2,.,qk1) = ?Pk i=1i ? Qk i=1(i) Qk1 i=1 qi1 i ? 1 Pk1 i=1 qi ?k1 . However, rather than needlessly complicate the presentation, we shall just write
3、the density as a function of the entire k-dimensional vector q. We also note that the constraint that P iqi= 1 forces the components of Q to be dependent. UWEETR-2010-00062 = 1,1,1 = .1,.1,.1 = 10,10,10 = 2,5,15 Figure 1: Density plots (blue = low, red = high) for the Dirichlet distribution over the
4、 probability simplex in R3for various values of the parameter . When = c,c,c for some c 0, the density is symmetric about the uniform pmf (which occurs in the middle of the simplex), and the special case = 1,1,1 shown in the top-left is the uniform distribution over the simplex. When 0 0 (k,)Gamma d
5、istribution with parameters k and D =AD=B means random variables A and B have the same distribution where (s) denotes the gamma function. The gamma function is a generalization of the factorial function: for s 0, (s + 1) = s(s), and for positive integers n, (n) = (n 1)! because (1) = 1. We denote th
6、e mean of a Dirichlet distribution as m = /0. Fig. 1 shows plots of the density of the Dirichlet distribution over the two-dimensional simplex in R3for a handful of values of the parameter vector . When = 1,1,1, the Dirichlet distribution reduces to the uniform distribution over the simplex (as a qu
7、ick exercise, check this using the density of the Dirichlet in (1).) When the components of are all greater than 1, the density is monomodal with its mode somewhere in the interior of the simplex, and when the components of are all less than 1, the density has sharp peaks almost at the vertices of t
8、he simplex. Note that the support of the Dirichlet is open and does not include the vertices or edge of the simplex, that is, no component of a pmf drawn from a Dirichlet will ever be zero. Fig. 2 shows plots of samples drawn IID from diff erent Dirichlet distributions. Table 2 summarizes some key p
9、roperties of the Dirichet distribution. When k = 2, the Dirichlet reduces to the Beta distribution. The Beta distribution Beta(,) is defi ned on (0,1) and has density f(x;,) = ( + ) ()()x 1(1 x)1. To make the connection clear, note that if X f3527d41efb383c74dad1180d061717Lff354488a165b9aefe3c2b530d
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