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 $Var(X) = {\sigma}^2$

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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• […] we look at an important concept that is an extension from Bayes Theorem, which we discussed […]