We look at the definitions first.

A continuous random variable, X, has a probability density function (PDF), f(\bullet) if f(x) \ge 0 and for all events A
P(X \in A) = \int_A f(y) dy
The CDF and PDF are related by F(x) = \int_{-\infty}^x f(y) dy
It is good to know that we have P(X \in (x - \frac{\epsilon}{2}, x + \frac{\epsilon}{2})) \approx \epsilon f(x)

We X has a normal distribution, X \sim N(\mu, {\sigma}^2), and f(x) = \frac{1}{\sqrt{2 \pi {\sigma}^2}}e^(-\frac{(x - \mu)^2}{2 {\sigma}^2}). And \mathbb{E}[X] = \mu while Var(X) = {\sigma}^2

Log normal Distribution Source: www.me.utexas.edu
Log normal Distribution Source: www.me.utexas.edu

We also have the log-normal distribution, X \sim LN (\mu, {\sigma}^2) and log(X) \sim N(\mu, {\sigma}^2). Here, \mathbb{E} [X] = e^(\mu + \frac{{\sigma}^2}{2}) and Var(X) = e^(2 \mu + {\sigma}^2) (e^{{\sigma}^2} - 1). The log-normal distribution is very important in financial applications, for starters, the Black Scholes Equation.

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