Explain the terms (i) probability mass function, (ii) probability density function and (iii) probability distribution function.

(i) Probability mass function

If X is a discrete random variable with distinct values x1, x2 , ..., xn , ..., then the function, denoted

by Pₓ (x) and defined by 

 This is defined to be the probability mass function or discrete probability function of X.

The probability mass function p(x) must satisfy the following conditions

(ii) Probability density function

The probability that a random variable X takes a value in the interval [t1,t2] (open or closed) is given by the integral of a function called the probability density function fX (x) :

Other names that are used instead of probability density function include density function, continuous probability function, integrating density function.

The probability density functions fX (x) or simply by f (x) must satisfy the following conditions.

(iii) Probability distribution function

If X is a continuous random variable with the probability density function fX (x), then the function FX (x) is defined by

is called the distribution function (d.f) or sometimes the cumulative distribution function (c.d.f) of the continuous random variable X .