Discrete and Random Variables

Difference Between Discrete and Continuous Random Variables

To understand difference between Discrete and Continuous variable we need to understand what a Random Variable means and how it is different than the normal algebric variable.

By Definition:

Random Variable: They are used to measure the outcome of a random process. It is just quantifying the outcome of random of process. This quantification is required to perform more mathematical analysis. They are usually represented by capital letters.

For eg.
X is a random variable that measures outcome of a toss of fair coin. We can assign value to X as:
X = { 1 when head otherwise 0 }

We can use this random variable and can calculate mathematical measures like what is the probability that we will get three Heads in 3 tosses of fair coin etc. Since tossing a fair coin is a pure random process.

This probability can be written as:
P(X =3 Head)

Non Random Variables: These are the variables that measures outcome of a non-random process. We can solve for these variables, we can calculate relationship among these variables and we can draw inferences about these variables. They are usually denoted in the small letters.

For eg:

Like x + 5 = 10 we can easily deduce x has one the value 5. If y = 2x + 3 this represents an equation of line and we can plot it and make conclusion is y is an linear additive function of x.

Random Variables can be further divided into "Continuous Random Variables" or "Discrete Random Variables". This classification is done basis on the measure or feature that a variable represents.

Discrete Random Variable: By meaning Discrete means "individually separate or distinct" while variable means something that can take different values at different times.  So a discrete variable is something that can take values from well defined finite set and these values are produced as outcome of a Random Process. However it can take values upto infinity. Say we have defined a set of all positive integer like S = {0,1,2,3,4.....,infinity } and a discrete variable X denotes all even numbers from the set S. So being a discrete variable X can take any value like 2,4,6,8,10 and so on. But X can not take any value in between 2 and 3, Since S is set of all positive integers. So X can not take any value between interval of 2 and 3. For example we have a data set of Name and DOB. Both the variables can take values from Set of all human Names and Set of all legitimate date of births. There is nothing like between 2 dates or 2 names. So these variables are discrete in nature.

In other words Discrete Random Variable represents an aspect of random process that can be count or that is countable however count can go up to infinity.

Continuous Random Variables:  These are the variables that can take all possible values between an interval of values generated as outcome of random process. In the above example variable Y (represents real numbers) is a continuous random variable when it can take any value between interval 2,3. That Y can have value 2.01,2.001,2.999999 etc. Another example would weight of all american males. So discrete random variables represents aspect of random process whose outcome are not countable. It is not possible to count all the all values that a continuous random variable represents.

To classify a random variable as a continuous variable or discrete variable we need to understand how measurement about those variable is taken. So prior to making this decision understands context and methodology with which measurement was taken. Suppose we are looking at the data set with weight of all males in India and while taking measurements weight is rounded off to nearest Kg. In this case variable weight became a discrete variable. However weight is taken on a high precision scale so that it can measure up to milligrams and so on it became a continuous variables.

Since Random variables are used for further mathematical analysis and the widely used technique is Probability Distributions. Both Continuous and Discrete Random Variables can be explored by different kind of Probability Distributions. Discrete Random Variables are usually expressed by Probability Mass Distribution and Continuous random variables are expressed in terms of Probability Density Function. We will take about them later. But their effectiveness is measured in terms of Sum of all the probabilities for probability mass function (Discrete Random Variable)  and it  less than or equal to 1. And for Random Continuous Variables its effectiveness is measured in terms of Probability Density Function's Area Under Curve (Integration of Probability Curve with limits from start of the curve to end of curve) and it is always less than or equal to 100%.

Good Luck !!!