Continuous Random Variable: Definition & Examples

Instructor: Sharon Linde

Sharon has a Masters of Science in Mathematics

How do you define a continuous random variable? This lesson will go over the definition and properties, and will give examples of this statistical concept.

Definition of Continuous Random Variable

Richard is struggling with his math homework today, which is the beginning of a section on random variables and the various forms these variables can take. He's read the opening pages of the section several times, but it's just not making any sense to him. He decides to call his grandfather, who used to teach high school math and is usually pretty good about explaining things in a way Richard can understand.

Grandpa Don tells Richard to read the definition that is in his book. Richard reads, 'A random variable can be defined as the numerical outcomes of random events.' Grandpa Don explains that this is just a fancy way of saying that anything that can be measured can have a random numerical result.

For example, the heights of all the students in Richard's class could be made into random variables if everyone's height was measured. There would be a value for the height of the shortest person and one for the tallest, and everyone else would fall somewhere in between those two extremes.

Types of Random Variables

Grandpa Don goes on to explain that there are two types of random variables: continuous and discrete. Richard responds, 'yeah, I think I get the discrete random variable, which takes on one of a very specific set of values. They have an example in the book about rolling a die. There are only 6 possible values that can come up: 1 through 6.'

Since Richard already has a handle on the discrete random variable, Grandpa Don switches to the continuous random variable. 'Right, which is different than the heights of your classmates we were just talking about. You can't ever get a roll of 1.5, but you could have someone that was between 5 feet 7 inches and 5 feet 8 inches tall. Since the height of anyone in the class can be anywhere on the continuum between the largest and smallest heights, we would call it a continuous random variable.'

Properties of Continuous Random Variables

Things are starting to make sense for Richard. There's one more important concept of continuous variables for him to learn, though - that no measurement of a continuous random variable can ever occur more than once. This means that no one in the entire world could be the same height as anyone in Richard's class.

What if there happen to be two people in his class, say identical twins, of the exact same height? Sounds tricky, but the reality is if it were possible to measure them accurately enough, they would not be exactly the same height. We might have to go out to 10 decimal places, or 100, but eventually there would be a slight difference.

If having 100 decimal places sounds impossible to you, you're right. It is impossible in actuality, but we're talking theory right now. It's theoretically possible to talk about someone's height to 100 decimal places, and if we did that there would not be any two people in the world that had the same height - even though there are over 7 billion people in the world (and counting!).

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