is
defined as the random variable vector, 
, the limit state function can divide the
design region into safe and failure regions along the boundary 
. Then, the probability of failure for a
system is
In reliability analysis, when a performance function or a
limit state function is
defined as the random variable vector, , the limit state function can divide the
design region into safe and failure regions along the boundary . Then, the probability of failure for a
system is
In which 
is a joint probability density
function for 
and 
is failure region. As
explained in the Monte Carlo simulation chapter has been widely used to
avoid the above multiple integration. It requires, however, too many
simulations. Thus, in this section, another approach is explained to estimate
the probability of failure, which is based on the uncorrelated standard normal
variable space.