What is domain and range of random variable?

Answer 1

Let S denote the sample space underlying a random experiment with elements

s 2 S. A random variable, X, is defined as a function X(s) whose domain is

S and whose range is a set of real numbers, i.e., X(s) 2 R1.

Example A: Consider the experiment of tossing a coin. The sample space is

S = fH; Tg. The function

X(s) =

½

1 if s = H

¡1 if s = T

is a random variable whose domain is S and range is f¡1; 1g.

Example B: Let the set of all real numbers between 0 and 1 be the sample

space, S. The function X(s) = 2s ¡ 1 is a random variable whose domain is S

and range is set of all real numbers between ¡1 and 1.

A discrete random variable is one whose range is a countable set. The

random variable defined in example A is a discrete randowm variable. A continuous

random variable is one whose range is not a countable set. The random

variable defined in Example B is a continiuos random varible. A mixed random

variable contains aspects of both these types. For example, let the set of

all real numbers between 0 and 1 be the sample space, S. The function

X(s) =

½

2s ¡ 1 if s 2 (0; 1

2 )

1 if s 2 [ 1

2 ; 1)

is a mixed random variable with domain S and range set that includes set of all

real numbers between ¡1 and 0 and the number 1.

Cummulative Distribution Function

Given a random variable X, let us consider the event fX · xg where x is any

real number. The probability of this event, i.e., Pr(X · x), is simply denoted

by FX(x) :

FX(x) = Pr(X(s) · x); x 2 R1:

The function FX(x) is called the probability or cumulative distribution

fuction (CDF). Note that this CDF is a function of both the outcomes of the

random experiment as embodied in X(s) and the particular scalar variable x.

The properties of CDF are as follows:

² Since FX(x) is a probability, its range is limited to the interval: 0 ·

FX(x) · 1.

² FX(x) is a non-decreasing function in x, i.e.,

x1 < x2 Ã! FX(x1) · FX(x2):

1

² FX(¡1) = 0 and FX(1) = 1.

² For continuous random variables, the CDF fX(x) is a unifromly continuous

function in x, i.e.,

lim

x!xo

FX(x) = FX(xo):

² For discrete random variables, the CDF is in general of the form:

FX(x) =

X

xi2X(s)

piu(x ¡ xi); x 2 R1;

where the sequence pi is called the probability mass function and u(x) is

the unit step function.

Probability Distribution Function

The derivative of the CDF FX(x), denoted as fX(x), is called the probability

density function (PDF) of the random variable X, i.e.

fX(x) = dF(x)

dx

; x 2 R1:

or, equivalently the CDF can be related to the PDF via:

FX(x) =

Z x

¡1

fX(u)du; x 2 R1:

Note that area under the PDF curve is unity, i.e.,

Z 1

¡1

fX(u)du = FX(1) ¡ FX(¡1) = 1 ¡ 0 = 1

In general the probability of a random variable X(s) taking values in the range

x 2 [a; b] is given by:

Pr(x 2 [a; b]) =

Z b

a

fX(x)dx = FX(b) ¡ FX(a):

For discrete random variables the PDF takes the general form:

fX(x) =

X

xi2X(s)

pi±(x ¡ xi):

Specifically for continuous random variables:

Pr(x = xo) = FX(x+

o ) ¡ FX(x¡o

) = 0:

2