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
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