# Log normal distribution

## Where do you meet this distribution?

- Finance, Economics : Change of stock price

## Shape of Distribution

### Basic Properties

- Two parameters are required (How can you get these).
- Continuous distribution defined on semi-bounded range
- This distribution is always asymmetric.

### Probability

- Cumulative distribution function
where

is cumulative distribution function of standard normal distribution. - Probability density function
where

is probability density function of standard normal distribution. - How to compute these on Excel.

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7

A B Data Description 0.5 Value for which you want the distribution 0.1 Value of parameter M 2 Value of parameter S Formula Description (Result) =NTLOGNORMDIST(A2,A3,A4,TRUE) Cumulative distribution function for the terms above =NTLOGNORMDIST(A2,A3,A4,FALSE) Probability density function for the terms above - Function reference : NTLOGNORMDIST

### Quantile

- Inverse function of cumulative distribution function
where

is cumulative distribution function of standard normal distribution. - How to compute this on Excel.

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A B Data Description 0.7 Probability associated with the distribution 0.1 Value of parameter M 2 Value of parameter S Formula Description (Result) =NTLOGNORMINV(A2,A3,A4) Inverse of the cumulative distribution function for the terms above

## Characteristics

### Mean – Where is the “center” of the distribution? (Definition)

- Mean of the distribution is given as
where

- How to compute this on Excel

1 2 3 4 5 A B Data Description 0.1 Value of parameter M 2 Value of parameter S Formula Description (Result) =NTLOGNORMMEAN(A2,A3) Mean of the distribution for the terms above - Function reference : NTLOGNORMMEAN

### Standard Deviation – How wide does the distribution spread? (Definition)

- Variance of the distribution is given as
where

Standard Deviation is a positive square root of Variance.

- How to compute this on Excel

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A B Data Description 0.1 Value of parameter M 2 Value of parameter S Formula Description (Result) =NTLOGNORMSTDEV(A2,A3) Standard deviation of the distribution for the terms above - Function reference : NTLOGNORMSTDEV

### Skewness – Which side is the distribution distorted into? (Definition)

- Skewness of the distribution is given as
where

- How to compute this on Excel

1 2 3 4 5 A B Data Description 0.1 Value of parameter M 2 Value of parameter S Formula Description (Result) =NTLOGNORMSKEW(A2,A3) Skewness of the distribution for the terms above - Function reference : NTLOGNORMSKEW

### Kurtosis – Sharp or Dull, consequently Fat Tail or Thin Tail (Definition)

- Kurtosis of the distribution is given as
where

- How to compute this on Excel

1 2 3 4 5 A B Data Description 0.1 Value of parameter M 2 Value of parameter S Formula Description (Result) =NTLOGNORMKURT(A2,A3) Kurtosis of the distribution for the terms above - Function reference : NTLOGNORMKURT

## Random Numbers

- Random number x is generated by inverse function method, which is for uniform random U,
where

is cumulative distribution function of standard normal distribution. - How to generate random numbers on Excel.

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A B Data Description 0.1 Value of parameter M 2 Value of parameter S Formula Description (Result) =NTRANDLOGNORM(100,A2,A3,0) 100 log normal deviates based on Mersenne-Twister algorithm for which the parameters above Note The formula in the example must be entered as an array formula. After copying the example to a blank worksheet, select the range A7:A106 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

- Function reference : NTRANDLOGNORM

## NtRand Functions

- If you already have parameters of the distribution
- Generating random numbers based on Mersenne Twister algorithm: NTRANDLOGNORM
- Computing probability : NTLOGNORMDIST
- Computing mean : NTLOGNORMMEAN
- Computing standard deviation : NTLOGNORMSTDEV
- Computing skewness : NTLOGNORMSKEW
- Computing kurtosis : NTLOGNORMKURT
- Computing moments above at once : NTLOGNORMMOM

- If you know mean and standard deviation of the distribution
- Estimating parameters of the distribution:NTLOGNORMPARAM

## Reference

- Wolfram Mathworld – Log Normal Distribution
- Wikipedia – Log-normal distribution
- Statistics Online Computational Resource