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

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?

• 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. – whuber Sep 18 at 15:09
• 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 $\sigma. \qquad$ – Michael Hardy Sep 18 at 15:22

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 $$\log X\sim N(\mu,\sigma^2)$$, then we say that the original distribution $$X$$ is lognormal with parameters $$\mu$$ and $$\sigma^2$$.

(And then the mean of $$X$$ is $$\exp\left(\mu+\frac{\sigma^2}{2}\right)$$, 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.)

• 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. – confused Sep 18 at 14:51
• Nice answer! I was just a few seconds late in posting mine. 😃 – Isabella Ghement Sep 18 at 14:59
• 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. – Michael Hardy Sep 18 at 15:22
• @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. – Stephan Kolassa Sep 18 at 15:24

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 = $$\sqrt{\exp[\sigma^2 - 1] \cdot \exp(2\mu + \sigma^2)}$$