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Hello I'm learning about probability distributions and this was the given definition for a lognormal distribution:

A random variable X follows a lognormal distribution if its natural logarithm, ln X, is normally distributed.

It then goes on to say:

The reverse is also true: If the natural logarithm of random variable X, ln X, is normally distributed, then X follows a lognormal distribution.

For the life of me I cannot make sense of the second part of this statement, it seems to imply there's an inverse to the original definition but I read them both to mean the same thing.

To me the first definition says: X is lognormally distributed if ln(X) is normally distributed. It then says:

The "reverse" also says the same thing to me: If ln(x) is normally distributed, then X is lognormal.

I feel like I'm missing something here, they specifically used the word "reverse" and I want to understand why.

If my question is unclear perhaps this will help: Reading that definition reminded of this:

If an object is a square then it is a rectangle. The reverse is not true, rectangles are not squares.

I understand this statement, I see how the original statement and the reverse are two different statements. I don't understand how the two statements above are not totally identical.

user3002540
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1 Answers1

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You are right - the two statements you listed are equivalent (which makes me wonder whether they are cited correctly).

Either way, the correct statements would be:

Statement 1: If $\ln(X)$ is normally distributed, then $X$ follows a lognormal distribution. I.e. $$\ln(X) \sim \mathcal{N}(\mu,\sigma^{2}) \Rightarrow X \sim Lognormal(\mu,\sigma^{2})$$

Statement 2: If $X$ follows a lognormal distribution, then $\ln(X)$ is normally distributed. I.e. $$X \sim Lognormal(\mu,\sigma^{2}) \Rightarrow \ln(X) \sim \mathcal{N}(\mu,\sigma^{2})$$

Or, since both of the above hold,

Statement 3: $X$ follows the lognormal distribution if and only if $\ln(X)$ is normally distributed. I.e. $$X \sim Lognormal(\mu,\sigma^{2}) \iff \ln(X) \sim \mathcal{N}(\mu,\sigma^{2})$$

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