The probability function P maps the events in the sample space of a random experiment to a number in the interval [0, 1]. The output P(A) of the function is the probability measure of an event A in the sample space S.

The probability function takes the form of probability mass function to represent the probabilities of a discrete random variable and probability density function which is used to determine the probabilities for a continuous random variable.
Probability function is thus the super set which includes all functions that are used for determining the probability under various contexts.

A Binomial experiment consists of repeated trials, each trial having only two possible outcomes.
The binomial probability function outputs the probabilities of the number of successes in a fixed number of trials.
The binomial formula for finding the probability of exactly X successes in n trials is given by
P(X : n,p) = C(n,X). px . qn-x.
Where X = 0,1,2,.......n.
Binomial probability function is a Probability mass function and
P(0) + P(1) + P(2) + ............P(n) = 1

In a Trinomial experiment, each trial has three possible outcomes.
If X a trinomial random variable consists of three mutually exclusive events E1, E2 and E3 with probabilities p1, p2 and p3 of occurring, and X1, X2 and X3 are the number of times the events E1, E2 and E3 will occur, then the probability X will occur is

P(X) = $\frac{n!}{X_{1}!X_{2}!X_{3}!}$$p_{1}^{X_{1}}p_{2}^{X_{2}}p_{3}^{X_{3}}$

A binomial probability formula involves considerable computation when n the number of trials is large. A Poisson variable is used as an approximation to its Binomial form when n is large, p is small and the independent occurrences take place over a period of time. The Poisson formula for computing the probability of exactly X successes,

P(X : λ) = $\frac{e^{-\lambda }\lambda ^{x}}{X!}$

The parameter λ used to describe a Poisson distribution is the mean of the random variables and λ = np.
Probability density functions are used to evaluate the probability of a continuous random variable.
A function f(x) is said to be the probability density function pdf of a continuous random variable X if
  1. f(x) ≥ 0 for all x
  2. $\int_{-\infty }^{\infty }f(x)dx$
Cumulative probability function gives the sum of probabilities for a discrete random variable X.
X = {x1, x2, x3,......xk,.......}
Cumulative distribution function F(X) is given by
F(x) = $\sum_{x_{i}<x}p(x_{i})$
F(xk) = $\sum_{i=1}^{k}p(x_{i})$
Cumulative distributive function cdf is used to compute the probability of a continuous random variable X for a given upper bound x.
F(x) = $\int_{-\infty }^{x}f(x)dx$
Joint probability functions are defined for two dimensional random variables as follows:

Discrete Case:
Let (X, Y) be a two dimensional discrete random variable. If p(xi, yj) be a real number ∈ [0, 1] associated with each of (xi, yj)
i, j = 1, 2, 3....., then pis called the joint probability function of (X, Y) if
  1. p(xi, yj) ≥ 0 for all i, j = 1, 2, 3......
  2. $\sum_{i=1}^{\infty }\sum_{j=1}^{\infty }p(x_{i},y_{j})$ = 1.

Continuous Random Variables:
The joint probability distribution in this case is generally presented in tabular form.Continuous Random Variables
Let (X, Y) be a two dimensional continuous random variable. Then a function f(x, y) is called the joint pdf of (X, Y) if

  1. f(x,y) ≥ 0 for all x,y
  2. $\int_{-\infty }^{\infty }\int_{-\infty }^{\infty }f(x,y)dxdy$ = 1.
Marginal probability distributions are found for multi-dimensional random variables by fixing the value of one of the variables as a constant.

Discrete Case:
If (X, Y) is a two dimensional discrete random variable with joint probability distribution p(xi, yj) then the marginal probability distributions for X and Y are defined as follows:
p(xi) = $\sum_{j}p(x_{i},y_{j})$
p(y) = $\sum_{i}p(x_{i},y_{j})$
In a table of joint probabilities p(xi) is the the sum of the ith row and p(yj) is the sum of the jth column.

Continuous Random Variable:
If (X,Y) is a two dimensional continuous random variable with pdf f(x, y) then the marginal distributions of X and Y are defined as follows:
g(x) = $\int_{-\infty }^{\infty }f(x,y)dy$
h(y) = $\int_{-\infty }^{\infty }f(x,y)dx$
Conditional probability functions are defined for two dimensional variables as follows:

Discrete case:
Suppose (X, Y) is a discrete two dimensional discrete variable with joint probability distribution p(xi, yj).
If p(xi) and q(yj) are marginal probability functions of X and Y respectively, then the conditional probability
distribution of Y on X is defined as

p($\frac{x}{y}$) = P(X = $\frac{x}{Y}$ = y) = $\frac{p(x_{i},y_{j})}{q(y_{j})}$

and the conditional probability distribution of X on Y is defined as
p($\frac{y}{x}$) = P(Y = $\frac{y}{X}$ = x) = $\frac{p(x_{i},y_{j})}{p(x_{i})}$

Continuous Case:
Suppose (X, Y) is a two dimensional continuous random variable with joint probability function f(x, y).

Then the conditional probability density function of X given that Y = y is defined by
g($\frac{x}{y}$) = $\frac{f(x,y)}{h(y)}$

and the conditional probability density function of Y given that X = x is defined by
h($\frac{y}{x}$) = $\frac{f(x,y)}{g(x)}$