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Monte Carlo Analysis
Monte Carlo Basic Features
Monte Carlo Basic Features
Monte Carlo analysis can be understood as a form of integration algorithm, or as an incremental
approach providing a gradual insight into the spread of distribution.
Monte Carlo as an Integration Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
Monte Carlo Incremental Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
Monte Carlo Analysis With a Varying Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
Monte Carlo as an Integration Algorithm
Monte Carlo analysis can be understood as a form of integration, where the expected value:
of a random variable Y = φ(H(X)) needs to be estimated. Depending on the chosen function φ,
several different quantities can be estimated:
• φ(h) = h
The mean of the output response Y = H(X) is estimated.
• φ(h) = h 2
The second order moment and the variance of the distribution are estimated.
• φ(h) = I {h ≥ y}
The probability of exceeding the threshold y is estimated.
The standard Monte Carlo approach (referred to as the “Monte Carlo method”, following
Metropolis and Ulam (1949)), is to use a sample X 1 , …, X n from the density f X to approximate
the expectation by the empirical average:
In a simple Monte Carlo analysis the values of the circuit characteristics are independent and
identically distributed (i.i.d.), therefore standard results apply (under some assumptions): the
Strong Law of Large Numbers and the Central Limit Theorem describe the convergence of the
sample mean. The average converges almost surely to the expectation E(H(X)) and the
speed of convergence can be assessed because the variance
Eldo® User's Manual, 15.3 417