4.1 Random Variables

Previously, we learned about probability experiments and the possible outcomes of the experiment (the sample space). For example, flipping a coin two times is an experiment, and the possible outcomes of that experiment are [latex]\{HH, HT, TH, TT\}[/latex]. We also learned how to calculate probabilities associated with the outcomes of the experiment. In the case of flipping a coin two times, the probability of getting two heads is [latex]25\%[/latex].

Instead of listing the outcomes of the experiment, we want to associate numerical values with the outcomes of an experiment. For the flipping the coin two times experiment, we can associate the number of heads we get on the two flips with the outcomes in the sample space:  two heads is associated with the outcome [latex]HH[/latex], one head is associated with the outcomes [latex]HT[/latex] and [latex]TH[/latex], and zero heads is associated with the outcome [latex]TT[/latex]. In this way, we can assign a numerical value to the outcomes of the experiment. This numerical description of the outcomes of an experiment is called a random variable.

A random variable is a numerical description of the outcomes of an experiment.   Each possible outcome of an experiment is associated with a numerical value based on the random variable. The numerical values the random variable takes on depends on how the variable is defined and the outcomes of the experiment. The values of a random variable can vary with each repetition of an experiment. Upper case letters such as [latex]X[/latex] or [latex]Y[/latex] denote a random variable. Lowercase letters like [latex]x[/latex] or [latex]y[/latex] denote the value of a random variable. If [latex]X[/latex] is a random variable, then [latex]X[/latex] is written in words, and [latex]x[/latex] is given as a number. For example, if [latex]X[/latex] is the number of children in a family, then [latex]x[/latex] represents a specific integer [latex]0, 1, 2, 3,....[/latex].

Variables in statistics differ from variables in intermediate algebra in the two following ways:

• The values a random variable can take on are not necessarily numbers. The values may be expressed in words. For example, if [latex]X[/latex] is hair colour, then the values of the random variable are {black, blond, grey, brown, red}.
• We can tell what specific value, [latex]x[/latex], the random variable [latex]X[/latex] takes only after performing the experiment.

Discrete Random Variables

A random variable that only takes on certain numerical values is called a discrete random variable. For example, the random variable defined as the number of heads obtained in two flips of a coin is a discrete random variable because the random variable can only take on the values [latex]0[/latex], [latex]1[/latex], and [latex]2[/latex].

Continuous Random Variables

A random variable that takes on any numerical value in an interval or a collection of intervals is called a continuous random variable. Examples of continuous random variables are random variables associated with measurements, such as time, weight, height, or distance. For example, the random variable defined as the height of students in a class is a continuous random variable because the height of a student may taken on any positive number.

Exercises

1. A baker is deciding how many batches of muffins to make to sell in his bakery. He wants to make enough to sell every one and no fewer.
   1. Define a random variable for this experiment.
   2. List the possible values of the random variable.
   3. Is the random variable discrete or continuous?
   Click to see Answer

   1. The number of batches of muffins the baker needs to make.
   2. [latex]0, 1, 2, 3, \ldots[/latex]
   3. Discrete.

    

2. A meteorologist monitors the temperature at the airport each day.
   1. Define a random variable for this experiment.
   2. List the possible values of the random variable.
   3. Is the random variable discrete or continuous?
   Click to see Answer

   1. The temperature at the airport on any given day.
   2. Any possible number.
   3. Continuous.

    

3. Ellen is supposed to practice her music three days a week.
   1. Define a random variable for this experiment.
   2. List the possible values of the random variable.
   3. Is the random variable discrete or continuous?
   Click to see Answer

   1. The number of times Ellen practices each week.
   2. [latex]0, 1, 2, 3[/latex]
   3. Discrete.

    

4. An IT helpdesk monitors the time a worker spends with a client on a call.
   1.  Define a random variable for this experiment.
   2. List the possible values of the random variable.
   3. Is the random variable discrete or continuous?
   Click to see Answer

   1. The amount of time, in minutes, the worker spends on a call.
   2. Any non-negative number.
   3. Continuous.

    

5. Javier volunteers in community events each month. He does not do more than five events in a month.
   1.  Define a random variable for this experiment.
   2. List the possible values of the random variable.
   3. Is the random variable discrete or continuous?
   Click to see Answer

   1. The number of events Javier does in a month.
   2. [latex]0, 1, 2, 3, 4, 5[/latex].
   3. Discrete.

    

6. A quality control expert monitors the volume of paint in 4-litre paint cans on an assembly line. The quality control expert randomly selects a paint can off of the assembly line to check its volume
   1. Define a random variable for this experiment.
   2. List the possible values of the random variable.
   3. Is the random variable discrete or continuous?
   Click to see Answer

   1. The number of litres of paint in the paint can.
   2. Any non-negative number.
   3. Continuous.

    

“4.1 Introduction to Discrete Random Variables“, “5.1 Introduction to Contiuous Random Varibles“, and “4.6 Exercises” from Introduction to Statistics by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.