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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