By Interrante L.V., Caspar L.A., Ellis A.B. (eds.)
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Extra resources for Materials Chemistry
F. of X is a function fX (x) satisfying P(X [ A) ¼ Á Á Á A f (X) dx ¼ Á Á Á A f (x1 , . . , xn ) dx1 Á Á Á dxn : Let gX (x) ¼ gX (x1 , x2 , . . , xn ) be a real-valued function deﬁned on the sample space of X. Then g(X) is a random variable and the expected value of g(X) is Eg(X) ¼ X 1 ð g(X)f (X) and Eg(X) ¼ X ð1 ÁÁÁ À1 g(X)f (X) dx À1 in the discrete and continuous cases, respectively. f. f. f. f. over all possible values of the other coordinates. For example, the marginal distribution of X1 is given by &P P Á Á Á xk f (x1 , x2 , .
V. X. Let mX and sX be the mean and standard deviation of X, mY and sY be the mean and standard deviation of Y, and Cov(X, Y) r ¼ Corr(X, Y) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Var(X) Var(Y) be the correlation coefﬁcient between X and Y. Then the bivariate normal probability density function of (X, Y) is given by & 1 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp À f (x, y) ¼ 2 2(1 À r 2) 2psx sy 1 À r " #) x À my 2 x À mx 2 x À mx x À my þ Â À 2r sx sx sy sy where r is the correlation between X and Y. 3 Dirichlet Distribution The Dirichlet distribution, often denoted Dir(a), is a family of continuous multivariate probability distributions parameterized by the vector a of positive real numbers.
It is important to realize that the covariance matrix must be allowed to be singular (thus not described by the above formula for which S21 is deﬁned). That case arises frequently in statistics. Note also that the Xi’s are in general not independent; they can be seen as the result of applying the linear transformation A to a collection of independent Gaussian variables Z. The distribution of a random vector X is a multivariate normal distribution that can be written in the following notation: X N(m, S) In the two-dimensional nonsingular case, the multivariate normal distribution reduces to the bivariate normal distribution.