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Can someone explain to me the parameters of a lognormal distribution?

Can someone explain to me the parameters of a lognormal distribution?

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7

$begingroup$

I'm doing some reading and this is the definition I got from DeGroot's book:

Does that mean the parameters are the same? For example, assume X is lognormally distributed and Y is normally distributed where Y = log(X). Is this saying that X and Y have the same mean and SDs even though they are different shaped distributions? If not, what distribution is μ and σ referring to?

In other words, if someone says X is lognormally distributed with mean μ and SD σ, do I need to do any conversion so that the mean and SD are in normal terms?

normal-distribution lognormal

share|cite|improve this question

edited 6 hours ago

confused

asked 9 hours ago

confusedconfused

6544 bronze badges

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Do not confuse parameters of a distribution family with moments. Although $mu,sigma$ parameterize the Lognormal distributions, they are not their means or standard deviations.
  $endgroup$
  – whuber♦
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  They have the same parameters, but they don't have the same mean or the same standard deviation. The two parameters, $mu$ and $sigma,$ that are the mean and standard deviation of $log X,$ are not the mean and standard deviation of $X.$ But the mean and standard deviation of $X$ are functions of $mu$ and $sigma. qquad$
  $endgroup$
  – Michael Hardy
  8 hours ago
  

  
  
  
  

add a comment
 | 

7

$begingroup$

I'm doing some reading and this is the definition I got from DeGroot's book:

Does that mean the parameters are the same? For example, assume X is lognormally distributed and Y is normally distributed where Y = log(X). Is this saying that X and Y have the same mean and SDs even though they are different shaped distributions? If not, what distribution is μ and σ referring to?

In other words, if someone says X is lognormally distributed with mean μ and SD σ, do I need to do any conversion so that the mean and SD are in normal terms?

normal-distribution lognormal

share|cite|improve this question

edited 6 hours ago

confused

asked 9 hours ago

confusedconfused

6544 bronze badges

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Do not confuse parameters of a distribution family with moments. Although $mu,sigma$ parameterize the Lognormal distributions, they are not their means or standard deviations.
  $endgroup$
  – whuber♦
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  They have the same parameters, but they don't have the same mean or the same standard deviation. The two parameters, $mu$ and $sigma,$ that are the mean and standard deviation of $log X,$ are not the mean and standard deviation of $X.$ But the mean and standard deviation of $X$ are functions of $mu$ and $sigma. qquad$
  $endgroup$
  – Michael Hardy
  8 hours ago
  

  
  
  
  

add a comment
 | 

$begingroup$

I'm doing some reading and this is the definition I got from DeGroot's book:

Does that mean the parameters are the same? For example, assume X is lognormally distributed and Y is normally distributed where Y = log(X). Is this saying that X and Y have the same mean and SDs even though they are different shaped distributions? If not, what distribution is μ and σ referring to?

In other words, if someone says X is lognormally distributed with mean μ and SD σ, do I need to do any conversion so that the mean and SD are in normal terms?

normal-distribution lognormal

share|cite|improve this question

edited 6 hours ago

confused

asked 9 hours ago

confusedconfused

6544 bronze badges

$endgroup$

share|cite|improve this question

edited 6 hours ago

confused

asked 9 hours ago

confusedconfused

6544 bronze badges

share|cite|improve this question

edited 6 hours ago

confused

asked 9 hours ago

confusedconfused

6544 bronze badges

asked 9 hours ago

confusedconfused

6544 bronze badges

asked 9 hours ago

confusedconfused

6544 bronze badges

2 Answers
2

active

oldest

votes

4

$begingroup$

assume X is lognormally distributed and Y is normally distributed where Y = log(X)

This is where you are confused. You don't make assumptions on two distributions, one of which just happens to be the log of the other.

Instead, you start with a distribution $X$. Then you consider $log X$. If $log Xsim N(mu,sigma^2)$, then we say that the original distribution $X$ is lognormal with parameters $mu$ and $sigma^2$.

(And then the mean of $X$ is $expleft(mu+fracsigma^22right)$, for instance, so the parameters are certainly not the same. This is also why it is better to speak of the "parameters" of a lognormal, rather than of the "mean and SD" - because it's very easy to get confused whether these refer to the actual mean or the log-mean, same for SD.)

share|cite|improve this answer

answered 9 hours ago

Stephan KolassaStephan Kolassa

56.5k1010 gold badges111111 silver badges208208 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ok thanks for clarifying. So generally when people provide the parameters, such as μ and σ, that refers to the distribution of Y or log(X). To get the mean of the lognormal distribution requires a conversion.
  $endgroup$
  – confused
  9 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Nice answer! I was just a few seconds late in posting mine. 😃
  $endgroup$
  – Isabella Ghement
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  But the parameters are the same. Then mean and the standard deviation and many other things are the same, but those two parameters are the same.
  $endgroup$
  – Michael Hardy
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MichaelHardy: yes, the parameters are the same, by definition. I just wince a bit every time someone calls $mu$ the "mean parameter of the lognormal", because it's only the log-mean, and it's so easy to confuse them.
  $endgroup$
  – Stephan Kolassa
  8 hours ago
  

  
  
  
  

add a comment
 | 

5

$begingroup$

Wikipedia has a nice article on log-normal distributions: https://en.m.wikipedia.org/wiki/Log-normal_distribution. The article reveals that the log-normally distributed X and the normally distributed log(X) have different means and standard deviations.

If X follows a log-normal distribution with parameters $mu$ and $sigma$, then $mu$ and $sigma$ represent the mean and standard deviation of the distribution of log(X), which is normal. In other words, the mean and standard deviation of the normally distributed log(X) are:

Mean of $log(X)=mu$

SD of $log(X) = sigma$

The mean and standard deviation of the log-normally distributed X are as follows:

Mean of X = $exp(mu + sigma^2/2)$

SD of X = $sqrtexp[sigma^2 - 1] cdot exp(2mu + sigma^2)$

share|cite|improve this answer

edited 8 hours ago

Michael Hardy

4,8241515 silver badges3131 bronze badges

answered 8 hours ago

Isabella GhementIsabella Ghement

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$endgroup$

add a comment
 | 

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4

$begingroup$

assume X is lognormally distributed and Y is normally distributed where Y = log(X)

This is where you are confused. You don't make assumptions on two distributions, one of which just happens to be the log of the other.

Instead, you start with a distribution $X$. Then you consider $log X$. If $log Xsim N(mu,sigma^2)$, then we say that the original distribution $X$ is lognormal with parameters $mu$ and $sigma^2$.

(And then the mean of $X$ is $expleft(mu+fracsigma^22right)$, for instance, so the parameters are certainly not the same. This is also why it is better to speak of the "parameters" of a lognormal, rather than of the "mean and SD" - because it's very easy to get confused whether these refer to the actual mean or the log-mean, same for SD.)

share|cite|improve this answer

answered 9 hours ago

Stephan KolassaStephan Kolassa

56.5k1010 gold badges111111 silver badges208208 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ok thanks for clarifying. So generally when people provide the parameters, such as μ and σ, that refers to the distribution of Y or log(X). To get the mean of the lognormal distribution requires a conversion.
  $endgroup$
  – confused
  9 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Nice answer! I was just a few seconds late in posting mine. 😃
  $endgroup$
  – Isabella Ghement
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  But the parameters are the same. Then mean and the standard deviation and many other things are the same, but those two parameters are the same.
  $endgroup$
  – Michael Hardy
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MichaelHardy: yes, the parameters are the same, by definition. I just wince a bit every time someone calls $mu$ the "mean parameter of the lognormal", because it's only the log-mean, and it's so easy to confuse them.
  $endgroup$
  – Stephan Kolassa
  8 hours ago
  

  
  
  
  

add a comment
 | 

4

$begingroup$

assume X is lognormally distributed and Y is normally distributed where Y = log(X)

This is where you are confused. You don't make assumptions on two distributions, one of which just happens to be the log of the other.

Instead, you start with a distribution $X$. Then you consider $log X$. If $log Xsim N(mu,sigma^2)$, then we say that the original distribution $X$ is lognormal with parameters $mu$ and $sigma^2$.

(And then the mean of $X$ is $expleft(mu+fracsigma^22right)$, for instance, so the parameters are certainly not the same. This is also why it is better to speak of the "parameters" of a lognormal, rather than of the "mean and SD" - because it's very easy to get confused whether these refer to the actual mean or the log-mean, same for SD.)

share|cite|improve this answer

answered 9 hours ago

Stephan KolassaStephan Kolassa

56.5k1010 gold badges111111 silver badges208208 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ok thanks for clarifying. So generally when people provide the parameters, such as μ and σ, that refers to the distribution of Y or log(X). To get the mean of the lognormal distribution requires a conversion.
  $endgroup$
  – confused
  9 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Nice answer! I was just a few seconds late in posting mine. 😃
  $endgroup$
  – Isabella Ghement
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  But the parameters are the same. Then mean and the standard deviation and many other things are the same, but those two parameters are the same.
  $endgroup$
  – Michael Hardy
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MichaelHardy: yes, the parameters are the same, by definition. I just wince a bit every time someone calls $mu$ the "mean parameter of the lognormal", because it's only the log-mean, and it's so easy to confuse them.
  $endgroup$
  – Stephan Kolassa
  8 hours ago
  

  
  
  
  

add a comment
 | 

$begingroup$

assume X is lognormally distributed and Y is normally distributed where Y = log(X)

This is where you are confused. You don't make assumptions on two distributions, one of which just happens to be the log of the other.

Instead, you start with a distribution $X$. Then you consider $log X$. If $log Xsim N(mu,sigma^2)$, then we say that the original distribution $X$ is lognormal with parameters $mu$ and $sigma^2$.

(And then the mean of $X$ is $expleft(mu+fracsigma^22right)$, for instance, so the parameters are certainly not the same. This is also why it is better to speak of the "parameters" of a lognormal, rather than of the "mean and SD" - because it's very easy to get confused whether these refer to the actual mean or the log-mean, same for SD.)

share|cite|improve this answer

answered 9 hours ago

Stephan KolassaStephan Kolassa

56.5k1010 gold badges111111 silver badges208208 bronze badges

$endgroup$

share|cite|improve this answer

answered 9 hours ago

Stephan KolassaStephan Kolassa

56.5k1010 gold badges111111 silver badges208208 bronze badges

answered 9 hours ago

Stephan KolassaStephan Kolassa

56.5k1010 gold badges111111 silver badges208208 bronze badges

answered 9 hours ago

Stephan KolassaStephan Kolassa

56.5k1010 gold badges111111 silver badges208208 bronze badges

5

$begingroup$

Wikipedia has a nice article on log-normal distributions: https://en.m.wikipedia.org/wiki/Log-normal_distribution. The article reveals that the log-normally distributed X and the normally distributed log(X) have different means and standard deviations.

If X follows a log-normal distribution with parameters $mu$ and $sigma$, then $mu$ and $sigma$ represent the mean and standard deviation of the distribution of log(X), which is normal. In other words, the mean and standard deviation of the normally distributed log(X) are:

Mean of $log(X)=mu$

SD of $log(X) = sigma$

The mean and standard deviation of the log-normally distributed X are as follows:

Mean of X = $exp(mu + sigma^2/2)$

SD of X = $sqrtexp[sigma^2 - 1] cdot exp(2mu + sigma^2)$

share|cite|improve this answer

edited 8 hours ago

Michael Hardy

4,8241515 silver badges3131 bronze badges

answered 8 hours ago

Isabella GhementIsabella Ghement

10.7k22 gold badges88 silver badges2727 bronze badges

$endgroup$

add a comment
 | 

5

$begingroup$

Wikipedia has a nice article on log-normal distributions: https://en.m.wikipedia.org/wiki/Log-normal_distribution. The article reveals that the log-normally distributed X and the normally distributed log(X) have different means and standard deviations.

If X follows a log-normal distribution with parameters $mu$ and $sigma$, then $mu$ and $sigma$ represent the mean and standard deviation of the distribution of log(X), which is normal. In other words, the mean and standard deviation of the normally distributed log(X) are:

Mean of $log(X)=mu$

SD of $log(X) = sigma$

The mean and standard deviation of the log-normally distributed X are as follows:

Mean of X = $exp(mu + sigma^2/2)$

SD of X = $sqrtexp[sigma^2 - 1] cdot exp(2mu + sigma^2)$

share|cite|improve this answer

edited 8 hours ago

Michael Hardy

4,8241515 silver badges3131 bronze badges

answered 8 hours ago

Isabella GhementIsabella Ghement

10.7k22 gold badges88 silver badges2727 bronze badges

$endgroup$

add a comment
 | 

$begingroup$

Wikipedia has a nice article on log-normal distributions: https://en.m.wikipedia.org/wiki/Log-normal_distribution. The article reveals that the log-normally distributed X and the normally distributed log(X) have different means and standard deviations.

If X follows a log-normal distribution with parameters $mu$ and $sigma$, then $mu$ and $sigma$ represent the mean and standard deviation of the distribution of log(X), which is normal. In other words, the mean and standard deviation of the normally distributed log(X) are:

Mean of $log(X)=mu$

SD of $log(X) = sigma$

The mean and standard deviation of the log-normally distributed X are as follows:

Mean of X = $exp(mu + sigma^2/2)$

SD of X = $sqrtexp[sigma^2 - 1] cdot exp(2mu + sigma^2)$

share|cite|improve this answer

edited 8 hours ago

Michael Hardy

4,8241515 silver badges3131 bronze badges

answered 8 hours ago

Isabella GhementIsabella Ghement

10.7k22 gold badges88 silver badges2727 bronze badges

$endgroup$

share|cite|improve this answer

edited 8 hours ago

Michael Hardy

4,8241515 silver badges3131 bronze badges

answered 8 hours ago

Isabella GhementIsabella Ghement

10.7k22 gold badges88 silver badges2727 bronze badges

edited 8 hours ago

Michael Hardy

4,8241515 silver badges3131 bronze badges

edited 8 hours ago

Michael Hardy

4,8241515 silver badges3131 bronze badges

answered 8 hours ago

Isabella GhementIsabella Ghement

10.7k22 gold badges88 silver badges2727 bronze badges

answered 8 hours ago

Isabella GhementIsabella Ghement

10.7k22 gold badges88 silver badges2727 bronze badges