Random Variables, Distribution function and Distribution Curve: A tutorial

What is  a random Variable?

Random Variables are a means of assigning numbers to outcomes of random processes or experiments or activities.

An example of random variable can be an “outcome of rolling a dice”. Let us denote this random variable by symbol “Odice-roll “. The values this random variable can take can be any integer between 1 and 6.

The Odice-roll is a discrete random-variable. Which means it can take only specific integer values like one among {1,2,3,4,5,6}. It cannot take non-integer values or values outside the 1 to 6 range.

An opposite of “discrete random variable” is a “continuous random variable”. A continuous random variable can take any real value (both integer and non-integer) in a given range. An example of continuous random variable can be “height of a person”. Lets represent it by symbol “Hperson“. The variable “Hperson” can take any value between 0 feet to 10 feet.

What is a Distribution Function?

A function which mathematically represents the outcomes from this random variable is called “Distribution”  or a “Distribution Function” of this random variable.

In the above case, the distribution function will be

Odice-roll = X ;   Where X belongs to {1,2,3,4,5,6}

What is a Distribution Curve?

A distribution curve is plotting of outcomes of the distribution function. It represents number of occurrences of each of the values.

Our distribution function for “outcomes from rolling dice” can be easily plotted on a 2-dimensional surface. Assuming that you rolled the dice a hundred times, and then you will see below distribution:

Distribution Curve for rolling a dice

As you can note, we don’t see a line in the graph but only points. This is because our function is a discrete one which can only take point values from 1 to 6.

If I were to plot a distribution curve for continuous random variable like “Hperson” or “Height of a person” you are likely to see a curve like below. As you can see, the curve is continuous implying that the height of a person can take any real value with in a reasonable range.

Continuous Random Variable Distribution - Heights of people

Hope this helps your understanding. Thank you for reading, please feel free to share your views and feedback through comments.

In the upcoming articles I will be explaining the concepts Mean, Standard-Deviation and Normal distribution.

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