Random variable

A random variable is a function that maps outcomes of a sample space to real numbers. It can be discrete or continous.

Distribution function

With every random variable X, we can associate a cumulative distribution function (cdf) $F_X(x)=P(X\le x)$.

$F_X(x)$ is a cdf iff:

1. $\underset{x\to 0}{lim}{F}_X(x)=0$ and $\underset{x\to \mathrm{\infty }}{lim}{F}_X(x)=1$
2. $F_X(x)$ is non-decreasing.
3. $F_X(x)$ is right-continuous.

Indentically distributed random variables

Random variables $X$ and $Y$ are identically distributed if $F_X(x)=F_Y(x)$ for all $x$.

Probability density (mass) function

The probability density function (pdf) or mass function (pmf) for a random variable $X$ is given by:

• $f_X(x)=P(X=x)$
• $f_X(x)=\frac{d}{dx}{F}_X(x)$ for continuous random variables.