Formal–statistical methods - Probabilistic Safety Assessment for Optimum Nuclear Power Plant Life Management (PLiM) - page 58

Random variable X is normally distributed and its standard deviation δ is
known.
In this case, with the reliability γ the upper limit of the error is

[2.3]

δ= δ t / n ,

where n is the number of tests (played values of X ); t is the argument of the
Laplace function for which Φ=g σ is the known standard deviation of X .

Random variable X is normally distributed, and its standard deviation σ
is unknown.
In this case, with reliability γ the upper limit of the error is

( ) / 2,

δ= tS n

g

/ , [2.4]

where n is the number of tests; s is the corrected standard deviation, t γ is
determined from tables.
Random variable X is distributed according to a law different from normal.
In this case, at a sufficiently large number of tests ( n > 30) with the
reliability which is approximately equal to γ, the upper limit of the error can
be calculated from formula [2.3], if the standard deviation σ of the random
variable X is known, but if σ is unknown, then we can substitute in the formula
[2.3] its estimate s - the 'corrected' standard deviation, or use the formula
[2/4]. Note that as n increases the difference between the results yielded by
both formulas becomes smaller. This is explained by the fact that the Student
distribution tends to normal at n →∞.
From the above it follows that the Monte Carlo method is closely connected
with problems of probability theory, mathematical statistics and computational
mathematics.In connection with the problem of simulation of random variables
(in particular, uniformly distributed) the methods of number theoryplay a
significant role.
Among other computational methods, the Monte Carlo method stands out
for its simplicity and generality.
It has some obvious advantages:
a) The method does not require any assumptions about the regularity, except
square integrability. This can be useful because very complex functions, whose
regularity properties are difficult to establish, are used quite often;
b) It leads to a feasible procedure even in the multidimensional case,
when numerical integration is not applicable, for example when the number
of measurements is greater than 10;
c) It is easy to apply for small restrictions and without a preliminary
analysis of the problem.
However, the methods have some disadvantages, namely:
a) The error bounds are not defined precisely and include some kind of
randomness. However, this is a more psychological than real difficulty;
b) The static error decreases slowly;
c) The need to have a sufficient number of random numbers; this is difficult

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