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,

$F(x_1, x_2,..,x_n)$ = $P(X_1\leq x_1,X_2\leq x_2,...,X_n\leq x_n)$