HP 12C: Log-Normal Distribution Parameter Conversions
Introduction
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 ) )
HP 12C Program: Log-Normal Distribution Parameter Conversions
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)
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)
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
Memory Registers:
R1 = μ
R2 = σ
R3 = E[x]
R4 = Var[x]
Source
"Log-normal distribution" Wikipedia. Last Edited May 18, 2023 and retrieved May 24, 2023. https://en.wikipedia.org/wiki/Log-normal_distribution
Eddie
All original content copyright, © 2011-2023. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.