# 3.10.2 Lognormal Distributions

###### 3.10.2 Lognormal Distributions

A random variable X is lognormally distributed if the natural logarithm of X is normally distributed. A lognormal distribution may be specified with its mean μ and variance σ2. Alternatively, it may be specified with the mean m and variance s2 of the normally distributed log X. We denote a lognormal distribution Λ(μ,σ2), but its PDF is most easily expressed in terms of m and s:

[3.100]

A lognormal distribution is illustrated in Exhibit 3.17.

Exhibit 3.17: The PDF of a lognormal distribution.

The expectation, standard deviation, skewness, and kurtosis of a lognormal distribution are, in terms of m and s,

[3.101]

[3.102]

[3.103]

[3.104]

If we know μ and σ instead of m and s, we can convert between these with

[3.105]

[3.106]

Formulas [3.101] and [3.102] provide the reverse conversion.

As with normal distributions, the CDF of a lognormal distribution exists but cannot be expressed in terms of standard functions. It can be valued using a standard normal table. Let X ~ Λ(μ,σ2) with corresponding parameters m and s. Then X = exp(sZ + m) for some Z ~ N(0,1). Denote the CDFs of X and Z as ΦX and ΦZ. By [3.95] and the definition of the lognormal distribution:

[3.107]

[3.108]

[3.109]

[3.110]

[3.111]

which can be looked up in a standard normal table. Note that step [3.108] depends critically on the monotonicity of the log function. Exhibit 3.18 shows how, a ≤ b if and only if log alog b.

Exhibit 3.18: Because the log function is monotone, a ≤ b if and only if log alog b.