Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Probability - Discrete probability distributions - Concept.
Discrete random variables.


 

The video on concepts accessed through the main Probability page reviewed the main concepts with examples of discrete random variables.

A summary of the main issues raised in that video is provided below:

 

Random:

The Macquarie dictionary defines random as being “not according to a pattern or a method”. Synonyms of random include "chance" and "undirected".

Using these ideas as a basis, the term random in probability and statistics implies that a selection of someone or something to be studied is not done with an intent or a deliberate purpose related to that person etc. Instead each person or object to be selected has a defined but unbiased probability or chance for being selected.


Hence the criterion of randomness aims to maximise the probability that a selected subset reflects more accurately the characteristics of the broader group or population. Randomness therefore minimises bias in the sample selection process.
With such a perspective, random implies made, done, happening or chosen without a pre-determined method or a conscious decision.

 

A variable:

In algebra, we are always using the term variable. It is a quantity generally represented by a letter or a symbol in an equation to describe a relationship amongst the “unknowns”. The value of one variable is written so that its depends on the value of another variable. If we use a function, then we can substitute any values for an independent variable and calculate the value of a dependent variable which was the subject of the equation.

This concept of a variable is different to the concept of a random variable in probability and statistics. In these fields of mathematics, the term is used in a more abstract and varied way.

 

A random variable:

A random variable is a variable having a probability distribution associated with it or a function that assigns a numerical value to each of a number of possible outcomes in an event space.


A one dimensional random variable consists of a finite set of values for a given variable (i.e. a characteristic to be investigated) and each of the values reflect the different probability of each aspect occurring or being selected. Unlike in the algebra context, the random variable in statistics has a set of values.


When rolling a dice, the characteristic investigated (so the random variable) is represented by (say) X which can take the values from 1 to 6.
Observing a random sequence of events, symbols or steps shows no pre-ordained order and does not follow a pre-determined pattern or combination. For the dice example, there is no sequence of numbersthrown which are known in advance (unless you are cheating ).

 

Discrete and continuous:

There are two types of random variables, discrete and continuous.

    1. A discrete variable is a variable whose value is obtained by counting the occurrences.
    2. A continuous variable is a variable whose value is obtained by measuring an underlying feature or characteristic.


A discrete random variable is a variable that represents numbers found by counting. For example:

  • the number of marbles in a jar;
  • the number of students present; or
  • number of heads when tossing two coins.

The probability distribution for each of these examples contains all the possible values of the relevant random variable and the probability associated with each outcome.

A continuous random variable might be a person's height, weight or age. The underlying scale in these circumstances cand be continually refined to make more and more precise measurements.