Probability density function

We say a function $f (x)$ is a probability density function if

\int_{- \infty}^{\infty} f (x) d x = 1.

We use the notion $P (a < X < b)$ to represent the probability of an event whenever $x$ is greater than $a$ or less than $b$ , and it is

P (a < X < b) = \int_{a}^{b} f (x) d x .

Cummulative distribution function

This function $P (a < X < x)$ is called cummulative distribution function and is denoted by $F_{X} (x)$ . Note that cummulative distribution function is an antiderivative of $f$ .

Expected value

The expected value or expactation, denoted as $E (x)$ , of a probability density function $f (x)$ is defined as the improper integral:

E (x) = \int_{- \infty}^{\infty} x \cdot f (x) d x

This value is analogous to an average and can also be interpreted as the “center of mass” in a certain context.

Variance

The variance of probability density function $f$ is

$\int_{- \infty}^{\infty} (x - m)^{2} f (x) d x$ ,

where $m$ is the expected value.