Tham khảo tài liệu 'simulation and the monte carlo method episode 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 10 PRELIMINARIES As a consequence of properties 2 and 7 for any sequence of independent random variables Xj . xn with variances ơ ị . Ơ2 Var a biXi 2X2 bnxn J ơỵ for any choice of constants a and bl . bn. For random vectors such as X Xi . Xn T it is convenient to write the expectations and covariances in vector notation. Definition Expectation Vector and Covariance Matrix For any random vector X we define the expectation vector as the vector of expectations M M1 . Mn T E X1 . E X T. The covariance matrix E is defined as the matrix whose Í j -th element is Cov Xj Xj E Xj - pi Xj - Mj . If we define the expectation of a vector matrix to be the vector matrix of expectations then we can write p E X and E E X-m X-m T Note that p and E take on the same role as p and Ơ2 in the one-dimensional case. Remark Note that any covariance matrix E is symmetric. In fact see Problem it is positive semidefinite that is for any column vector u UT E lO 0 . FUNCTIONS OF RANDOM VARIABLES Suppose Xi . xn are measurements of a random experiment. Often we are only interested in certain functions of the measurements rather than the individual measurements themselves. We give a number of examples. EXAMPLE Let X be a continuous random variable with pdf ỈX and let z aX b where a f 0. We wish to determine the pdf fz of X. Suppose that a 0. We have for any Fz z P Z SỈ z p x z - 6 a Fx z - 6 a . Differentiating this with respect to z gives fz z fxifz b a a. For a 0 we similarly obtain fz z x z fi a a .Thus in general fz z fx . a a FUNCTIONS OF RANDOM VARIABLES 11 EXAMPLE Generalizing the previous example suppose that z p X for some monotonically increasing function g. To find the pdf of z from that of X we first write Fz z P Z o p x Fx where p-1 is the inverse of g. Differentiating with respect to z now gives For monotonically decreasing functions jjp-1 z in the first equation needs to be replaced with its negative value. EXAMPLE Order Statistics Let Xi .
