Probability density function
We say a function $f(x)$ is a probability density function if
\[\int_{-\infty}^\infty f(x)dx=1.\]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
\[P(a<X<b) = \int_a^bf(x)dx.\]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:
\[E(x) = \int_{-\infty}^\infty x \cdot f(x) dx\]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
\(\int_{-\infty}^\infty (x-m)^2f(x)dx\),
where $m$ is the expected value.