(12C) Log-Normal Distribution Parameter Conversions
05-29-2023, 12:41 AM
Post: #1
 Eddie W. Shore Senior Member Posts: 1,585 Joined: Dec 2013
(12C) Log-Normal Distribution Parameter Conversions
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
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