Evaluating dependability metrics of critical systems: Monte ... - iaria

Suppose now that we sample each pair (Y k ,Z k ) with the same
uniform r.v. U k : Y k = F −1
Y (U k) and Z k = F −1
Z (U k).
Using the fact that F −1 −1
Y
and FZ
are non increasing, we can easily
prove that Cov(Y k ,Z k ) ≥ 0.
This means that if we define a new estimator ˜X n as
we have
and
˜X n = 1 n
n∑ [
F
−1
Y (U k) − F −1
Z (U k) ]
k=1
E(˜X n ) = E(¯X n ) = µ,
V(˜X n ) ≤ V(¯X n ).
This technique is called common variables in Monte Carlo theory.
G. Rubino (INRIA) Monte Carlo DEPEND 2010 21 / 72