PENELOPE 2003 - OECD Nuclear Energy Agency

1.2. Random sampling methods 9
Numerical inverse transform
The inverse transform method can also be efficiently used for random sampling from
continuous distributions p(x) that are given in numerical form, or that are too complicated
to be sampled analytically. To apply this method, the cumulative distribution
function P(x) has to be evaluated at the points x i of a certain grid. The sampling
equation P(x) = ξ can then be solved by inverse interpolation, i.e. by interpolating in
the table (ξ i ,x i ), where ξ i ≡ P(x i ) (ξ is regarded as the independent variable). Care
must be exercised to make sure that the numerical integration and interpolation do not
introduce significant errors.
(a)
(b)
p (x)
p (x)
x
x
Figure 1.2: Random sampling from a continuous distribution p(x) using the numerical inverse
transform method with N = 20 intervals. a) Piecewise constant approximation. b) Piecewise
linear approximation.
A simple, general, approximate method for numerical sampling from continuous
distributions is the following. The values x n (n = 0, 1, . . . , N) of x for which the
cumulative distribution function has the values n/N,
P(x n ) =
∫ xn
x min
p(x) dx = n N , (1.37)
are previously computed and stored in memory. Notice that the exact probability of
having x in the interval (x n , x n+1 ) is 1/N. We can now sample x by linear interpolation: