Probability distributions are of two types: discrete and continuous. The outcome of the throw of a dice is discrete while distribution of weights of a group of people is continuous. A cumulative distributive function $P(x)$ is the probability of getting a value less than a certain value $x$. For example, if a student is chosen randomly from a class of $50$, what is the probability of choosing a kid with weight less than $40$ kg? This can be solved using cumulative distribution function.
Definition
If $p(x)$ is the density function given for a probability distribution, then the cumulative distributive function $P(x)$ is calculated as the integral of $p(x)$ over the interval $-\infty$ to $x$. The cumulative distributive function of $p(x)$,
$F(x) = \int_{-\infty }^{x}p(t)$For example: A probability distributive function is given as $f(x)$ = $\frac{x^2}{3}$ for $0\ <\ x\ <\ 2$, then what will be the cumulative distribution function?
We need to integrate the probability distributive function from $0$ to $x$.
$F(x)$ = $\int_{0}^{x}p(t)$ = $\int_{0}^{x}$
$\frac{x^2}{3}$ =
$\frac{x^3}{9}$ $|_{0}^{x}$ =
$\frac{x^3}{9}$
Formula
The cumulative distributive function of a continuous random value $X$ can be defined as:
$F(X)$ = $\int_{-\infty }^{x}p(t)$ for $-\infty\ <\ x\ <\ \infty$
For a discrete random value X, the cumulative distributive function can be defined as:
$F(X) = \sum_{t<x}^{ } p(t)$
Properties
The cumulative distribution function for a density function $p(x)$ gives:
1) The part of the given population which has value less than $x$.
2) The probability of getting the value less than $x$.Properties of cumulative distribution function $F(x)$:
1) $F(x)$ goes to zero as $x$ tends to minus infinity.
2) $F(x)$ tends to $1$ as $x$ tends to positive infinity.
3) $F(x)$ is non-decreasing.
Multivariate Case
The bivariate cdf of two variables X and Y is defined as,
$F(x, y)$ = $P(X\leq x,Y\leq y)$
The multivariate cumulative distribution function for several variables $X_1, X_2,.. X_n$ can be defined as,
$F(x_1, x_2,..,x_n)$ = $P(X_1\leq x_1,X_2\leq x_2,...,X_n\leq x_n)$