Notation

Multivariate probability density function (continuous random variables):

\[ f_X(x) \]

where \(X\) is a vector of \(n\) random variables:

\[ X=[X_1, X_2, \ldots , X_n] \]

realizations of a random variable \(X_i\) are denoted with lower-case letters: \(x_i\).

Note

No distinction is made between a random vector (a vector of random variables) and a single random variables; both use notation \(X\). You will have to use the context of the problem to determine which is the case (which should be obvious, if there is more than one random variable considered!).

Functions of Random Variables

A generic function of interest will usually be represented with letter \(q\):

\[ Q=q_X(x) \]

where upper-case \(Q\) implies that the output of the function is also a random variable (perhaps also a vectorized output). If the function \(q_X\) includes non-random (deterministic) parameters, they will be denoted using \(\theta\) to distinguish from random variables:

\[ Q=q_X(x,\theta) \]

Note

Similar to random vectors, deterministic parameters may also be represented as vectors (this helps with linear algebra computations, for example, sensitivity calculation). For the case where \(q\) has \(m\) deterministic parameters:

\[ \theta = [\theta_1, \theta_2, \ldots, \theta_m] \]

Limit-State Function

The letter \(g\) is used for the limit-state function; a special case of a function of random variables, \(q\), formulated such that failure is explicitly defined as the situation \(g_X(x)<0\).

\[ p_f = \int_\Omega g_X(x) \; \mathrm{d}X \]

In this case, \(\Omega\) is the failure region: the set of all \(x\) such that the component described by \(q_X(x)\) and \(g_X(x)\) fails. If \(G\) is the random variable output of \(g_X(x)\), then the failure probability can also be found as follows:

\[ p_f = P[g<0] = F_g(g=0) = \int_{-\infty}^{0} f_G(g) \; \mathrm{d}G \]

Unfortunately the integral can rarely be evaluated directly due to \(g_X(x)\) being a non-linear in \(X\) and/or the distribution of \(G\) being non-Gaussian.

Note

The use of capital \(G\) here deviates from that of the Der Kiureghian textbook, where \(G_U(u)\) is used to denote the limit-state function in the standard normal space (i.e., \(g_U(u)\) in this book).