# Understanding the difference between normal distribution and lognormal distribution

I'm having trouble understanding the difference between a normal distribution and lognormal distribution. Here's what I've done so far.

Definitions of lognormal curves: "A continuous distribution in which the logarithm of a variable has a normal distribution"

2) Lognormal Distribution

Important property of this distribution is that it does not take values less than 0. But how do we get this shape? A lognormal distribution is very much what the name suggest "lognormal". I explain this as follows: Imagine that you have a function that is the exponent of some input variable. The input variable itself is a normal distribution function.

e.g. $y=k e^{x}$

How can an input be the normal distribution?

I understand that if you plug in an input, square it, and then take the log of it, it's the normal distribution. But how does this tie into the lognormal distribution?

• The easiest way to understand and memorize this concept is as follows: a random variable follows a lognormal distribution if its $\textbf{log}\text{arithm}$ follows a normal distribution. Commented Sep 17, 2013 at 17:36

When any random variable $x$ has an distribution $X$ then any function y=$f(x)$ will have generally speaking a different distribution $Y$ deducted from $X$. You confuse distributions and variables. So the positively valued $x$ has log-normal distribution iff the new variable $y=\ln(x)$ has a normal distribution $N(\mu,\sigma)$. Log-normal distribution's density itself has an analytical form: $$Y(x;\mu,\sigma) = \frac{1}{x \sigma \sqrt{2 \pi}}\, e^{-\frac{(\ln x - \mu)^2}{2\sigma^2}},\ \ x>0$$ where $\mu$ and $\sigma^2$ respectively mean and variance of the corresponding normal distribution see change-of-variables rule.

• ??? God i suck at math.... Commented Sep 18, 2013 at 14:17
• Can we get an example??? Commented Nov 21, 2013 at 2:48
• Here's an example using the fact that stock prices are modeled using the lognormal distribution if the return has a normal distribution: financetrain.com/… Commented Aug 25, 2017 at 0:06