(12C) Log-Normal Distribution Parameter Conversions

[Eddie W. Shore]

The log-normal distribution is transformation of a standard normal variable, where for a standard normal variable t, then a random variable x follows a log-normal distribution, with the form:

x = e^(μ + t * σ)

where:

μ = mean
σ = standard deviation (sample)

The distribution takes the positive values of x. The cumulative distributive function of the log-normal distribution (the area between 0 and x) is:

pdf = 1/2 * (1 + erf((ln x - μ) ÷ (σ * √2)) )

erf is the error function.

erf(θ) = 2 ÷ √(π) * ∫(e^(-s^2) ds, s = 0 to s = θ)


This program on today's blog focuses on the relationship between the distribution mean (μ), standard deviation (σ), the arithmetic expected value (E[x]), and the arithmetic variance (Var[x]):

E[x] =e^(μ + σ^2 ÷ 2)

Var[x] = (e^(σ^2) - 1) * e^(2 * μ + σ^2)

μ = ln( E[x]^2 ÷ √(Var[x] + E[x]^2) )

σ = √( ln (1 + Var[x] ÷ E[x]^2 ) )

Calculate E[x] and Var[x] from μ and σ

Instructions:

To find E[x] and Var[x]:
1. Store μ in memory register 1
2. Store σ in memory register 2
3. Run the program. E[x] is shown in the X stack and is stored in memory register 3. Var[x] is shown in the Y stack in memory register 4.

Code:
(Step: Key Code: Key)
(assume program starts with step 00)

Code:

01:  45, 2:   RCL  2
02:  2:    2
03:  21:  y^x
04:  44, 0:   STO 0
05:  43, 22:  e^x
06:  1:  1
07:  30:  -
08:  2:  2
09:  45, 1:  RCL 1
10:  20:  ×
11:  45, 0:  RCL 0
12:  40:  +
13:  43, 22:  e^x
14:  20:  ×
15:  44, 4:  STO 4
16:  45, 0:  RCL 0
17:  2:   2
18:  10:  ÷
19:  45, 1:  RCL 1
20:  40:   +
21:  43, 22:  e^x
22:  44, 3:  STO 3
23:  44, 33, 00:  GTO 00

Lines 01 to 03: Store σ^2 in memory register 0


Examples (answers are rounded to four decimal places):

Example 1
Inputs: μ = 1, σ = 0.5
Results: E[x] = 3.0802, Var[x] = 2.6948

Example 2
Inputs: μ = 0, σ = 1
Results: E[x] = 1.6487, Var[x] = 4.6708


Calculate μ and σ from E[x] and Var[x]


Instructions

To find μ and σ:
1. Store E[x] in memory register 3
2. Store Var[x] in memory register 4
3. Run the program. μ is shown in the X stack and is stored in memory register 1. σ is shown in the Y stack in memory register 2.

Code:
(Step: Key Code: Key)
(assume program starts with step 00)

Code:

01:  45, 4:  RCL 4
02:  45, 3:  RCL 3
03:  2:   2
04:  21:  y^x
05:  44, 0:  STO 0
06:  10:  ÷
07:  1:  1
08:  40:  +
09:  43, 23:  LN
10:  43, 21:  √
11:  44, 2:  STO 2
12:  45, 0:  RCL 0
13:  45, 0:  RCL 0
14:  45, 4:  RCL 4
15:  40:  +
16:  43, 21:  √
17:  10:  ÷
18:  43, 23:  LN
19:  44, 1:  STO 1
20:  43, 33, 00:  GTO 00

Lines 01 to 03: Store E[x]^2 in memory register 0
Lines 12 to 13: Put two copies of memory register 0 on to the stack

Examples (answers are rounded to four decimal places):

Example 1
Inputs: E[x] = 1.84, Var[x] = 0.36
Results: μ = 0.5592, σ = 0.3180

Example 2
Inputs: E[x] = 5.03, Var[x] = 1.72
Results: μ = 1.5825, σ = 0.2565

"Log-normal distribution" Wikipedia. Last Edited May 18, 2023 and retrieved May 24, 2023. https://en.wikipedia.org/wiki/Log-normal_distribution