Discrete distribution

Discrete distribution

Random variable of discrete type

A SAMPLE SPACE S may be difficult to describe if the elements of S are not numbers. Let discuss how one can use a rule by which each simple outcome of a random experiment, an element s of S , may be associated with a real number x .

DEFINITION OF RANDOM VARIABLE

It may be that the set S has elements that are themselves real numbers. In such an instance we could write $X\left(s\right)=s$ so that X is the identity function and the space of X is also S . This is illustrated in the example below.

We can recognize two major difficulties:

• In many practical situations the probabilities assigned to the event are unknown.
• Since there are many ways of defining a function X on S , which function do we want to use?

Let X denotes a random variable with one-dimensional space R , a subset of the real numbers. Suppose that the space R contains a countable number of points; that is, R contains either a finite number of points or the points of R can be put into a one-to- one correspondence with the positive integers. Such set R is called a set of discrete points or simply a discrete sample space .

Furthermore, the random variable X is called a random variable of the discrete type , and X is said to have a distribution of the discrete type . For a random variable X of the discrete type, the probability $P\left(X=x\right)$ is frequently denoted by f(x) , and is called the probability density function and it is abbreviated p.d.f. .

Let f(x) be the p.d.f. of the random variable X of the discrete type, and let R be the space of X . Since, $f\left(x\right)=P\left(X=x\right)$ , x belongs to R , f(x) must be positive for x belongs to R and we want all these probabilities to add to 1 because each $P\left(X=x\right)$ represents the fraction of times x can be expected to occur. Moreover, to determine the probability associated with the event $A\subset R$ , one would sum the probabilities of the x values in A .

That is, we want f(x) to satisfy the properties

• $P\left(X=x\right)$ ,
• ${\displaystyle \sum _{x\in R}f\left(x\right)=1;}$
• $P\left(X\in A\right)={\displaystyle \sum _{x\in A}f\left(x\right)}$ , where $A\subset R.$

Usually let $f\left(x\right)=0$ when $x\notin R$ and thus the domain of f(x) is the set of real numbers. When we define the p.d.f. of f(x) and do not say zero elsewhere, then we tacitly mean that f(x) has been defined at all x’s in space R , and it is assumed that $f\left(x\right)=0$ elsewhere, namely, $f\left(x\right)=0$ , $x\notin R$ . Since the probability $P\left(X=x\right)=f\left(x\right)>0$ when $x\in R$ and since R contains all the probabilities associated with X , R is sometimes referred to as the support of X as well as the space of X .