Log normal distribution (from X α β α β α - William & Mary
[Pages:2]Log normal distribution (from ) The shorthand X log normal(, ) is used to indicate that the random variable X has the log normal distribution with parameters and . A log normal random variable X with parameters
and has probability density function
f (x) =
1
e-
1 2
(ln(x/)/)2
x 2
x>0
for and > 0. The log normal distribution can be used to model the lifetime of an object, the weight of a person, or a service time. The central limit theorem indicates that the log normal distribution is useful for modeling random variables that can be thought of as a product of several independent random variables. The probability density function with three different parameter settings is illustrated below.
f (x)
1.0
0.8 = 5, = 0.5
0.6
0.4
= 1, = 1
0.2
= 1, = 2
0.0
x
024
6
8 10
The cumulative distribution function on the support of X is
F (x)
=
P(X
x)
=
1 2
+
1 2
erf
2 (ln (x) - ) 2
x > 0,
where
erf(x)
=
2
x e-t2 dt.
0
The survivor function on the support of X is
S(x) = P(X x) = 1 - 1 erf 22
2 (ln (x) - ) 2
x > 0.
The hazard function on the support of X is
h(x)
=
f (x) S(x)
=
-2e-(ln(x)-)2/22
1
x-1-1
-1 + erf
2 (ln (x) - ) 2
-1
1
x > 0.
The cumulative hazard function on the support of X is
H(x) = - ln S(x) = ln (2) + i - ln -1 + erf
2 (ln (x) - ) 2
x > 0.
The inverse distribution function, moment generating function, and characteristic function of X are mathematically intractable.
The median of X is .
The population mean, variance, skewness, and kurtosis of X are
E[X ] = e2/2
V [X ] = 2e2(e2 - 1)
E
X -?
3
= (e2 + 2)(e2 - 1)1/2
E
X -?
4
= e42 + 2e32 + 3e22 - 3
APPL verification: The APPL statements
assume(alpha > 0); assume(beta > 0); X := [[x-> (1/(x*beta*sqrt(2*Pi)))*exp((-1/2)*(ln(x/alpha)/beta)^2)],
[0,infinity],["Continuous","PDF"]]; CDF(X); SF(X); HF(X); CHF(X); Mean(X); Variance(X); Skewness(X); Kurtosis(X);
verify the cumulative distribution function, survivor function, hazard function, cumulative hazard function, population mean, variance, skewness, and kurtosis.
2
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