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Concept and Definitions of Random Variables

1. Introduction

This is the first lecture on random variables. In this lecture, the basic concept of random

variable, its definition, different types of random variables, their probability distribution and

properties etc. are discussed.

2. Concept of Random Variable

Concept of Random Variable (RV) is most important to the probability theory and itsapplications. It is critical to always remember that: a Random Variable is not a variable.

According to classical concept, a random variable is a function denoted by or , that

map each points (outcomes of an experiment) over a sample space to a numerical value on

the real line. The Fig. 1. provides the visualization of this concept.

Fig. 1. Random Variable

Notations: Generally a Random Variable is denoted as, uppercase letter and its specific

values are denoted as lowercase letters. Thus, or denotes the RV and x denotes a

specific value of it.

Examples:

a. Number of rainy days in a month. Suppose, it is denoted by D . Some value of D, say

5, indicates 5 rainy days are observed in a particular month. Here D is the random

variable and d = 5.

b. Number of road accidents along a particular stretch of a road in a year. Suppose, it is

denoted by X . Some value of X , such as, x = 10 indicates 10 road accidents over that

stretch of road has occurred in a particular year.

Sample space

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c. Strength of concrete. Suppose it is denoted by C . Some value of C , such as,

c = 25 N/mm2, indicates 25 N/mm2 strength was observed for particular sample of

concrete.

3. Definition of Random Variable (RV)

Random Variable: A random variable ‘X’ is a process to assign a number to every

outcome of a random experiment. The resulting function must satisfy following two

conditions –

a. the set is an event for every .

b. the probabilities of the events and are equal to zero.

If not explicitly stated, all random variables are real.

4. Types of Random Variables (RVs)

There can be three possible types of RVs depending on the possible set of values.

4.1. Discrete Random Variable: If the possible set of values that a Random Variable is

assigned with some probability is finite, the random variable is called as Discrete Random

Variable. For Discrete Random Variable, all the probabilities, assigned at specific values, are

greater than or equal to zero. Summation of these probabilities for all possible values is

equal to 1.

Example 1. The number of rainy days at a particular location over a period of one month.

Example 2. Number of road accidents over a particular stretch of a national highway during a

year.

Example 3. Traffic volume at a particular section of a road.

4.2. Continuous Random Variable: If the possible set of values is a range (not discrete

values) over which the probability of a Random Variable is defined, then the random variable

is called as Continuous Random Variable. For Continuous Random Variable, probability of

any specific value is zero, however the probability of any infinitesimally small set of values isgreater than or equal to zero and the integration of the entire area under the curve on the

random variable axis is equal to 1.

Example 1. The amount of rain received at a particular place over a period of one year.

Example 2. Compressive strength of a concrete cube.

4.3. Mixed Random Variable: If some range of possible set of values of a Random Variable is

discrete and for the other range of possible set is continuous, then the random variable is

called as Mixed Random Variable.

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Example 1. Depth of rainfall at a particular rainguage station. There can be some fixed

probability for zero values, which can be defined by a concentrated probability value. For

other range (greater than zero), probability can be defined by a continuous probability density

function. Thus, the random variable for records of depth of rainfall is a mixed type.

5. Special Type of Random Variables

There is a special type of Random Variable beyond the three defined before. It deals with

occurrence of event following a particular condition. This is known as, Indicator Random

Variable. It is a special kind of Random Variable associated with occurrence of an event. An

Indicator Random Variable, I A represents all the outcomes in the set A as 1 and all the

outcomes outside A are 0.

6. Independence of Random Variables

Two Random Variables, R1 and R 2 are independent if for all , there exist thecondition that, .

Random Variables R1 , R2 , R3 ,..., Rt are mutually independent if for all , there

exist the condition that, .

A collection of Random Variables are said to k-wise independent if all the subsets of k-

variables are mutually independent .

When the number of variables reduced to 2 and if the subsets are mutually independent , the

random variables are said to be pair-wise independent.

7. Probability Distribution of Random Variables

The distribution function of Random Variable, X is the function for any x

between and . In the subsequent lectures, we will differentiate between probability

density function (pdf) and cumulative distribution function (CDF). In this section we mean

CDF by Probability Distribution.

7.1. General notation: Distribution function of X , Y , and are denoted by

, , respectively. Variables x ,y, z (inside the parenthesis can be denoted by

any letter.

This is also known as, Cumulative Distribution Function (CDF). This will be discussed along

with probability density function (pdf) later.

7.2. Probability Distribution of different Random Variables

Probability distributions for various types of Random Variable are classified following their

properties.

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7.2.1. Discrete Probability Distribution: It is a mathematical function (denoted as p(x) ) that

satisfies the following properties:

a. The probability of any event x can take a specific value p(x) , mathematically denoted as,

.

b. p(x) is non-negative for all real .

c. The sum of p(x) over all possible values of x is 1.

Though mathematically there is no restriction, in practice, discrete probability distribution

function is defined only for integer values.

7.2.2. Continuous Probability Distribution: It is a mathematical function (denoted as F x(x) )

that satisfies the following properties:

a. For all x,

b. It is monotonically increasing continuous function

c. It is 1 at and 0 at , i.e. and .

7.2.3. Mixed Probability Distribution: It is mathematical function (denoted as F x(x)) that

satisfies the following properties:

a. For all x,

b. It is monotonically increasing continuous function with sudden jumps or steps

c. It is 1 at and 0 at , i.e. and .

7.3. Usage in Statistics

There are several applications of concept of Random Variable and its probability distribution

in the field of statistics, viz. to calculate intervals for parameters and to calculate critical

regions, to determine reasonable distributional model for univariate data, to verify

distributional assumptions, to study the simulation of random numbers generated from a

specific probability distribution etc.

7.4. Usage in Civil Engineering

In Civil Engineering, concept of Random Variable and its probability distribution are used in

numerous applications. Here are some examples in different subfields of Civil Engineering:

in Water Resources Engineering: Analysis of flood frequency; Next in Geotechnical

Engineering: Distribution of in-situ stresses in rock surrounding an opening, uniaxial

compressive strength etc.; then in Structural Engineering: Distribution of damage stress in

masonry structure etc. to check seismic vulnerability; further in Transportation Engineering:

Traffic volume analysis etc.

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7.5. Percentiles of Random Variables

The u-percentile of a Random Variable is the smallest number xu so that

. Where, xu is the inverse of the distribution function Fx (x) , i.e.

within the domain and the range of being .

7.6. Properties of Distribution Functions

Notations: , where .

Property 1. , .

We can prove this property in the following way:

and .

Property 2. F(X) is a non-descending function of X , if x1< x2, then .


Since and for some , is a subset of the event .

Hence . Here F(X) increases from 0 to 1 as x increases from to

.

Property 3. If , then for any .


Since (from previous proof), and suppose that for every . Again,

since is an impossible event. So, for each .

Property 4.


The events and are mutually exclusive and . So,

, then , .

Property 5.The function F(x) is continuous from the right: F(x+)=F(x).


Since, and when as

.

Property 6.

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and are mutually exclusive and

again, . So,

or

.

Property 7.


Putting and in Property 6 , we get,

. Now taking ,

Property 8.


Using Property 6 and 7 , we get, and since

and are mutually exclusive events,

or

or

8. Concluding Remarks

Random Variable is not a variable, rather a function which maps all the feasible outcome of

an experiment on the real line or a set of real numbers. Random Variables can be either

discrete or continuous if the set of events defined by variable is either finite or infinite

numbers respectively. Mixed Random Variables are the combination of the both. It is also

true for probability distributions of them. Concept of Random Variable and its distributions

have numerous specific applications in Civil Engineering related problems. These problems

will be discussed in following lectures.

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