Probability Mass Function

Probability Mass Function(and probability distribution of a discrete random variable) 

Suppose X is a one-dimensional discrete random variable taking at most a countably infinite number of values  X₁>  X₂, '" With each possible outcome  X_i , , we aSsociate a number Pi = P ( X =  X_i ) = p ( X_i ). called the probability of Xi. The numbers p ( X_i); i:; 1,2,.,.. must satisfy the following conditions:
(i) p ( x≥ ∀ i,
(ii) 1: p (  X_i )= 1

This function p is called the probability mass function of the random variable X and the set ( X_i, p (Xi) ) is called the probability distribution (p.d.) of the r.v. X.
Remarks:

1. The set of values which X takes is called the spectrum of the random variable. 
2. For discrete random- variable, a knowledge of the probability mass function enables us to compute probabilities of arbitrary events. In fact, if E is a set of real numbers, we have

P ( X ϵ E) = Σ _xϵE∩SP(x), where S is the sample space.

Illustration. 

Toss of coin, S = {H.T}. Let X be the random variable defined by 

X (H) = I, i.e., X = I, if 'Head' occurs. 

X ( T) = 0, i.e., X = 0, if  'Tail' occurs. 

If the coin is 'fair' the probability function is given by 

P( {H} )=P( {T} )=1/2

and we can speak. of the probability,distribution of the random variable X as 

P(X= I)=P( {H} )=1/2

P(X=0)=P( (T) = 1/2