- Examples of discrete variables
- Discrete variables and continuous variables
- Solved problems of discrete variables
- -Solved exercise 1
- Solution
- -Solved exercise 2
- Solution
- Probability distributions
- References
A discrete variable is a numerical variable that can only assume certain values. Its distinctive feature is that they are countable, for example the number of children and cars in a family, the petals of a flower, the money in an account and the pages of a book.
The objective of defining variables is to obtain information about a system whose characteristics can change. And since the number of variables is enormous, establishing what type of variables it is with allows to extract this information in an optimal way.
The number of petals on a daisy is a discrete variable. Source: Pixabay.
Let's analyze a typical example of a discrete variable, from among those already mentioned: the number of children in a family. It is a variable that can take on values such as 0, 1, 2, 3, and so on.
Note that between each of these values, for example between 1 and 2, or between 2 and 3, the variable admits none, since the number of children is a natural number. You cannot have 2.25 children, therefore between the value 2 and the value 3, the variable called "number of children" does not assume any value.
Examples of discrete variables
The list of discrete variables is quite long, both in different branches of Science and in everyday life. Here are a few examples that illustrate this fact:
-Number of goals scored by a certain player throughout the season.
-Money saved in pennies.
-The energy levels in an atom.
-How many clients are served in a pharmacy.
-How many copper wires does an electrical cable have.
-The rings on a tree.
-Number of students in a classroom.
-Number of cows on a farm.
-How many planets does a solar system have?
-The number of light bulbs that a factory produces during a given hour.
-How many pets does a family have?
Discrete variables and continuous variables
The concept of discrete variables is much clearer when compared with that of continuous variables, which are the opposite since they can assume countless values. An example of a continuous variable is the height of students in a physics class. Or its weight.
Let us suppose that in a college the shortest student is 1.6345 m and the tallest 1.8567 m. Certainly, between the heights of all the other students, values will be obtained that fall anywhere in this interval. And since there is no restriction in this regard, the variable "height" is considered continuous in that interval.
Given the nature of discrete variables, one might think that they can only take their values in the set of natural numbers or at most in that of integers.
Many discrete variables take integer values frequently, hence the belief that decimal values are not allowed. However, there are discrete variables whose value is decimal, the important thing is that the values assumed by the variable are countable or countable (see resolved exercise 2)
Both discrete and continuous variables belong to the category of quantitative variables, which are necessarily expressed by numerical values with which to perform various arithmetic operations.
Solved problems of discrete variables
-Solved exercise 1
Two unloaded dice are rolled and the values obtained on the upper faces are added. Is the result a discrete variable? Justify your answer.
Solution
When two dice are added, the following outcomes are possible:
In total there are 11 possible outcomes. As these can only take the specified values and not others, the sum of the roll of two dice is a discrete variable.
-Solved exercise 2
For quality control in a screw factory an inspection is carried out and 100 screws are randomly chosen in a batch. The variable F is defined as the fraction of defective screws found, where f is the values that F is taking. Is it a discrete or continuous variable? Justify your answer.
Solution
To answer, it is necessary to examine all the possible values that f can have, let's see what they are:
Probabilities of each are: p (X = x i) = {1/6, 1/6, 1/6, 1/6, 1/6, 1/6}
Figure 2. The roll of a die is a discrete random variable, Source: Pixabay.
The variables in solved exercises 1 and 2 are discrete random variables. In the case of the sum of the two dice, it is possible to calculate the probability of each of the numbered events. For defective screws, more information is required.
Probability distributions
A probability distribution is any:
-Table
-Expression
-Formula
-Graph
That shows the values that the random variable takes (either discrete or continuous) and their respective probability. In any case, it must be observed that:
Where p i is the probability that the i-th event occurs and it is always greater than or equal to 0. Well, the sum of the probabilities of all events must be equal to 1. In the case of rolling the dice, add all the values of the set p (X = x i) and easily check that this is true.
References
- Dinov, Ivo. Discrete Random Variables and Probability Distributions. Retrieved from: stat.ucla.edu
- Discrete and Continuous Random Variables. Retrieved from: ocw.mit.edu
- Discrete Random Variables and Probability Distributions. Retrieved from:
- Mendenhall, W. 1978. Statistics for Management and Economics. Grupo Editorial Ibearoamericana. 103-106.
- Random Variables Problems and Probability Models. Recovered from: ugr.es.