log_normalPDF

function log_normalPDF(x, mu, sigma)

Description:

Calculates the probability density function
$$
f(x) = frac{1}{sqrt{2pitildesigma^2}x} cdot expleft[-frac{1}{2}cdot left(frac{log(x)-tilde mu}{tildesigma}right)^2right]
$$
of the log-normal distribution with mean (mu) and variance (sigma^2) using
$$
tilde sigma^2 = logleft(1 + frac{sigma^2}{mu^2}right) quad text{and} quad tilde mu = log(mu) – frac{tilde sigma^2}{2}.
$$

Input arguments:

• real*8 :: x
  The point where to evaluate the probability density function of the log-normal distribution.

Optional arguments:

• real*8 :: mu
  The mean of the distribution. If not present, the function uses (mu = e^{0.5}). Note that this input variable needs to be strictly greater than zero.
• real*8 :: sigma
  The variance (sigma^2) of the distribution. If not present, the function uses (sigma^2 = ecdot (e-1)). Note that this input variable needs to be strictly greater than zero.

Return Value:

• real*8 :: log_normalPDF
  The value of the density at x.

References

• For further reading refer to:
  • Toral, R. & Colet, P. (2014). Stochastic Numerical Methods: An Introduction for Students and Scientists. Weinheim: Wiley.
• This routine is used in the following programs:
  • prog02_15.f90