Multivariate Probability Distributions

Multivariate probability distributions are extensions of univariate probability distributions that jointly account for more than one random variable. Bivari­ate and trivariate distributions are special cases where two and three random variables, respectively, are involved. The fundamental basis of multivariate probability distributions is described in Sec. 2.3.2. In general, the availability of multivariate distribution models is significantly less than that for univari­ate cases. Owing to their frequent use in multivariate modeling and reliabil­ity analysis, two multivariate distributions, namely, multivariate normal and multivariate lognormal, are presented in this section. Treatments of some mul­tivariate nonnormal random variables are described in Secs. 4.5 and 7.5. For other types of multivariate distributions, readers are referred to Johnson and Kotz (1976) and Johnson (1987).

Several ways can be used to construct a multivariate distribution (Johnson and Kotz, 1976; Hutchinson and Lai, 1990). Based on the joint distribution dis­cussed in Sec. 2.2.2, the straightforward way of deriving a joint PDF involving K multivariate random variables is to extend Eq. (2.19) as

f x(x) = f 1(X1) X f 2(X2 | X1) X—X fK(X1, X2, . . ., Xk-1) (2.105)

in which x = (x1, x2,…, xK)t is a vector containing variates of K random vari­ables with the superscript t indicating the transpose of a matrix or vector. Ap­plying Eq. (2.105) requires knowledge of the conditional PDFs of the random variables, which may not be easily obtainable.

One simple way of constructing ajoint PDF of two random variables is by mix­ing. Morgenstern (1956) suggested that the joint CDF of two random variables could be formulated, according to their respective marginal CDFs, as

F 1,2(xb x2) = Fhx1)F2(x2){1 + в[1 — F 1(x1)][1 — F2(x2)]} for -1 < в < 1

(2.106)

in which Fk(xk) is the marginal CDF of the random variable Xk, and в is a weighting constant. When the two random variables are independent, the weighting constant в = 0. Furthermore, the sign of в indicates the positive­ness or negativeness of the correlation between the two random variables. This equation was later extended by Farlie (1960) to

F1,2(xb x2) = F1U1)F2(x2)[1 + ef 1U1)f2(x2)] for-1 < в < 1 (2.107)

in which fk (xk) is the marginal PDF of the random variable Xk. Once the joint CDF is obtained, the joint PDF can be derived according to Eq. (2.15a).

Constructing a bivariate PDF by the mixing technique is simple because it only requires knowledge about the marginal distributions of the involved random variables. However, it should be pointed out that the joint distribu­tion obtained from Eq. (2.106) or Eq. (2.107) does not necessarily cover the entire range of the correlation coefficient [-1, 1] for the two random variables under consideration. This is illustrated in Example 2.20. Liu and Der Kiureghian (1986) derived the range of the valid correlation coefficient value for the bivariate distribution by mixing, according to Eq. (2.106), from various combinations of marginal PDFs, and the results are shown in Table 2.4.

Nataf (1962), Mardia (1970a, 1970b), and Vale and Maurelli (1983) proposed other ways to construct a bivariate distribution for any pair of random variables. This was done by finding the transforms Zk = t(Xk), for k = 1, 2, such that Z1 and Z2 are standard normal random variables. Then a bivariate normal distri­bution is ascribed to Z1 and Z2. One such transformation is zk = Ф-1[^(xk)], for k = 1, 2. A detailed description of such a normal transformation is given in Sec. 4.5.3.

Example 2.20 Consider two correlated random variables X and Y, each ofwhich has a marginal PDF of an exponential distribution type as

fx(x) = e—x for x > 0 fy(y) = e—y for y > 0

To derive a joint distribution for X and Y, one could apply the Morgenstern formula. The marginal CDFs of Xand Y can be obtained easily as

Fx(x) = 1 — e—x for x > 0 Fy(y) = 1 — e—y for y > 0

According to Eq. (2.106), the joint CDF of X and Y can be expressed as

Fx, y(x, y) = (1 — e—x)(1 — e—y )(1 + вe—x—y) for x, y > 0

Then the joint PDF of X and Y can be obtained, according to Eq. (2.7a), as

fx, y(x, y) = e—x—y [1 + в(2e—x — 1)(2e—y — 1)] for x, y > 0

TABLE 2.4 Valid Range of Correlation Coefficients for the Bivariate Distribution Using the Morgenstern Formula Marginal distribution N U SE SR T1L T1S LN GM T2L T3S N 0.318 U 0.326 0.333 SE 0.282 0.289 0.25 SR 0.316 0.324 0.28 0.314 T1L 0.305 0.312 0.27 0.303 0.292 T1S 0.305 0.312 0.27 0.303 0.292 0.292 LN <0.318 <0.326 <0.282 <0.316 <0.305 <0.305 <0.318 GM <0.318 <0.326 <0.282 <0.316 <0.305 <0.305 <0.318 <0.381 T2L <0.305 <0.312 <0.270 <0.303 < 0.292 < 0.292 <0.305 <0.305 < 0.292 T3S <0.305 <0.312 <0.270 <0.303 < 0.292 < 0.292 <0.305 <0.305 < 0.292 < 0.292 NOTE: N = normal; U = uniform; SE = shifted exponential; SR = shifted Rayleigh; T1L = type Ilargest value; T1S = type I smallest value; LN = lognormal; GM = gamma; T2L = type II largest value; T3S = type III smallest value. SOURCE : After Lin and Der Kiureghian (1986).

To compute the correlation coefficient between X and Y, one first computes the covari­ance of X and Y as Cov( X, Y) = E (XY) — E (X) E (Y ),in which E (XY) is computed by

f ж f ж a

E (XY) = xyfx, y(x, y) dxdy = 1 +-

J0 J0 4

Referring to Eq. (2.79), since the exponential random variables X and Y currently considered are the special cases of в = 1, therefore, xx = Xy = 1 and ax = ay = 1. Conse­quently, the covariance of X and Y is 0/4, and the corresponding correlation coefficient is 0/4. Note that the weighing constant 0 lies between [-1, 1]. The preceding bivariate exponential distribution obtained from the Morgenstern formula could only be valid for X and Y having a correlation coefficient in the range [-1/4, 1/4].