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?
Best Answer
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.