# Discrete and continuous random variables | Probability and Statistics | Khan Academy (2023)

## Introduction

Defining discrete and continuous random variables. Working through examples of both discrete and continuous random variables.

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## Content

What.

We're going to see in this video is that random variables come in two varieties.

You have discrete random variables, and you have continuous random variables.

And, discrete random variables.

These are essentially random variables that can take on distinct or separate values.

And, we'll give examples of that in a second.

So that comes straight from the meaning of the word, discrete in the English language-- distinct or separate values., While, continuous-- and I guess just another definition for the word discrete in the English language would be polite, or not obnoxious, or kind of subtle.

That is not what we're talking.

We are not talking about random variables that are polite.

We're talking about ones that can take on distinct, values., And continuous, random variables.

They can take on any value in a range.

And.

That range could even be infinite.

So any value in an interval.

So.

With those two definitions out of the way.

Let's look at some actual random variable.

Definitions.

And I want to think together about whether you would classify them as discrete or continuous random, variables.

So.

Let's say that.

I have a random variable capital, X.

And.

It is equal, to-- well.

This is one that we covered in the last video.

It's 1.

If my fair coin is heads.

It's 0, if my fair coin is tails.

So, is this a discrete or a continuous random variable? Well,? This random variable right over here can take on distinctive values.

It can take on either a 1 or it could take on a 0.

Another way to think about it.

Is you can count the number of different values it can take? On.

This is the first value it can take on.

This is the second value that it can take on.

So.

This is clearly a discrete random variable.

Let's define random variable Y as equal to the mass of a random animal selected at the New Orleans zoo, where I grew up.

The Audubon Zoo.

Y is the mass of a random animal selected at the New Orleans zoo.

Is, this a discrete random variable or a continuous random variable? Well.

The exact mass-- and I should probably put that qualifier here.

I'll even add it here, just to make it really, really clear.

The, exact mass of a random animal, or a random object in our universe.

It can take on any of a whole set of values.

I mean, who knows exactly the exact number of electrons that are part of that object right at that moment? Who knows the neutrons, the protons, the exact number of molecules in that object, or a part of that animal.

Exactly at that, moment? So that mass, for example, at the zoo, it might take on a value anywhere, between-- well, maybe close to 0.

There's, no animal that has 0 mass.

But.

It could be close to zero, if we're thinking about an ant, or we're thinking about a dust mite, or maybe, if you consider even a bacterium, an animal.

I believe bacterium is the singular of bacteria.

And.

It could go all the way.

Maybe.

The most massive animal in the zoo is the elephant of some kind.

And I, don't know what it would be in kilograms, but it would be fairly large.

So.

Maybe you can get up all the way to 3,000, kilograms, or probably larger.

Let's, say 5,000, kilograms.

I, don't know what the mass of a very heavy elephant-- or a very massive elephant, I should say--.

Actually is.

It may be something fun for you to look at.

But, any animal could have a mass anywhere in between here.

It does not take on discrete values.

You could have an animal that is exactly maybe 123.75921 kilograms.

And, even there, that actually might not be the exact mass.

You might have to get even more precise, --10732.

0, 7, And I.

Think you get the picture.

Even, though this is the way I've defined it now, a finite interval.

You can take on any value in between here.

They are not discrete values.

So.

This one is clearly a continuous random variable.

Let's say that random variable Y, instead of it being this.

Let's say it's the year that a random student in the class was born.

Is, this a discrete or a continuous random variable? Well, that year, you literally, can define it as a specific, discrete year.

It.

Could be 1992, or it could be 1985, or it could be.

2001.

There are discrete values that this random variable can actually take.

On.

It won't be able to take on any value.

Between, say, 2000 and 2001.

It'll either be 2000 or it'll, be 2001 or 2002.

Once again.

You can count the values it can take.

On.

Most of the times that you're dealing with, as in the case right, here, a discrete random variable--.

Let me make it clear this one over here is also a discrete random variable.

Most of the time that you're dealing with a discrete random, variable, you're, probably going to be dealing with a finite number of values.

But.

It does not have to be a finite number of values.

You can actually have an infinite potential number of values that it could take on-- as long as the values are countable.

As long as you can literally say, OK.

This is the first value it could take.

On, the second, the third.

And.

You might be counting forever, but as long as you can literally list-- and it could be even an infinite list.

But.

If you can list the values that it could take, on, then you're dealing with a discrete random variable.

Notice.

In this scenario, with the zoo, you could not list all of the possible masses.

You could not even count them.

You might attempt to-- and it's a fun exercise to try at least once, to try to list all of the values this might take.

On.

You might say, OK.

Maybe it could take on 0.01 and maybe 0.02.

But wait.

You just skipped an infinite number of values that it could take on, because it could have taken on 0.011, 0.012.

And.

Even between those, there's, an infinite number of values.

It could take on.

There's, no way for you to count the number of values that a continuous random variable can take.

On.

There's no way for you to list them.

With, a discrete random variable.

You can count the values.

You can list the values.

Let's do another example.

Let's, let random variable, Z, capital Z, be the number ants born tomorrow in the universe.

Now, you're, probably arguing that there aren't ants on other planets.

Or.

Maybe there are ant-like creatures, but they're not going to be ants as we define them.

But.

How do we know? So number of ants born in the universe.

Maybe? Some ants have figured out interstellar travel of some kind.

So, the number of ants born tomorrow in the universe.

That's, my random variable, Z.

Is, this a discrete random variable or a continuous random variable? Well.

Once again, we can count the number of values this could take, on.

This could be 1., It could be 2., It could be 3., It could be 4., It could be 5, quadrillion, ants.

It could be 5, quadrillion and 1.

We can actually count the values.

Those values are discrete.

So once again.

This right over here is a discrete random.

Variable.

This is fun.

So let's keep doing more of these.

Let's say that I have random variable.

X.

So we're not using this definition, anymore.

Now I'm going to define random variable X to be the winning time--.

Now let me write it.

This way.

The exact winning time for the men's 100-meter dash at the 2016 Olympics.

So, the exact time that it took for the winner-- who's, probably going to be Usain Bolt, but it might not be.

Actually, he's aging a little bit.

But, whatever the exact winning time for the men's 100-meter in the 2016 Olympics.

And, not the one that you necessarily see on the clock.

The exact, the precise time that you would see at the men's, 100-meter dash.

Is this a discrete or a continuous random variable? Well, the way I've defined, and this one's a little bit.

Tricky.

Because, you might say it's countable.

You, might say, well.

It could either be 956, 9.56, seconds, or 9.57 seconds, or 9.58 seconds.

And.

You might be tempted to believe that, because when you watch the 100-meter dash at the Olympics, they measure it to the nearest hundredths.

They round to the nearest hundredth.

That's, how precise their timing, is.

But I'm talking about the exact winning time, the exact number of seconds it takes for that person, to, from the starting gun, to cross the finish line.

And there.

It can take on any value.

It can take on any value, between-- well, I, guess they're limited by the speed of light.

But.

It could take on any value you could imagine.

It might be anywhere between 5 seconds and maybe 12 seconds.

And.

It could be anywhere in between there.

It might not be 9.57.

That might be what the clock says, but in reality the exact winning time could be 9.571, or it could be 9.572359.

I.

Think you see what I'm saying.

The exact precise time could be any value in an interval.

So.

This right over here is a continuous random variable.

Now.

What would be the case, instead of saying the exact winning time, if instead I defined X, to be the winning time of the men's 100 meter dash at the 2016 Olympics, rounded to the nearest hundredth? Is, this a discrete or a continuous random variable? So? Let me delete this.

I've changed the random variable now.

Is, this going to be a discrete or a continuous random variable? Well now.

We can actually count the actual values that this random variable can take.

On.

It might be 9.56.

It could be 9.57., It could be 9.58.

We can actually list them.

So in this case.

When we round it to the nearest hundredth, we can actually list of values.

We are now dealing with a discrete random, variable., Anyway, I'll.

Let you go there.

Hopefully.

This gives you a sense of the distinction between discrete and continuous random variables.

## FAQs

### What is the difference between discrete and continuous random variables Khan Academy? ›

For example, the outcome of rolling a die is a discrete random variable, as it can only land on one of six possible numbers. Continuous random variables, on the other hand, can take on any value in a given interval.

How do you know whether a random variable is continuous or discrete? ›

A random variable is called discrete if its possible values form a finite or countable set. A random variable is called continuous if its possible values contain a whole interval of numbers.

Can a random variable be both discrete and continuous? ›

A random variable can be either discrete (having specific values) or continuous (any value in a continuous range). The use of random variables is most common in probability and statistics, where they are used to quantify outcomes of random occurrences.

Is the a discrete random variable a continuous random variable or not a random variable? ›

A discrete variable is a variable whose value is obtained by counting. A continuous variable is a variable whose value is obtained by measuring. A random variable is a variable whose value is a numerical outcome of a random phenomenon. A discrete random variable X has a countable number of possible values.

Is discrete math easier than continuous math? ›

In discrete mathematics, you will (early on) have to work with induction to do proofs on sequences and series. This can be unintuitive for some poeple. However, all that said and done, learning finite-discrete mathematics is easier than either.

Can you give 5 examples of discrete random variables? ›

Examples of discrete random variables include the number of children in a family, the Friday night attendance at a cinema, the number of patients in a doctor's surgery, the number of defective light bulbs in a box of ten.

Can you give 5 examples of continuous random variables? ›

In general, quantities such as pressure, height, mass, weight, density, volume, temperature, and distance are examples of continuous random variables.

What is an example of a discrete and continuous random variable? ›

For example, if X is equal to the number of miles (to the nearest mile) you drive to work, then X is a discrete random variable. You count the miles. If X is the distance you drive to work, then you measure values of X and X is a continuous random variable.

Is Age discrete or continuous? ›

The exact age is a continuous variable, but age is often rounded down to the closest integer.

What are the two conditions for a continuous random variable? ›

A random variable X is continuous if possible values comprise either a single interval on the number line or a union of disjoint intervals.

### Does a continuous random variable have a probability? ›

Unlike the case of discrete random variables, for a continuous random variable any single outcome has probability zero of occurring.

How do you prove two continuous random variables are independent? ›

Independent Random Variables: When two jointly continuous random variables are independent, we must have fX|Y(x|y)=fX(x). That is, knowing the value of Y does not change the PDF of X. Since fX|Y(x|y)=fXY(x,y)fY(y), we conclude that for two independent continuous random variables we must have fXY(x,y)=fX(x)fY(y).

What is an example of not a continuous random variable? ›

Therefore, number of cars in a street is not a continuous variable.

What is an example of a discrete random variable? ›

A discrete random variable is a variable that can take on a finite number of distinct values. For example, the number of children in a family can be represented using a discrete random variable.

Is time A discrete or continuous variable? ›

Examples of continuous data include weight, height, length, time, and temperature.

Do you need to be good at calculus for discrete math? ›

Calculus isn't really needed to understand discrete math, but if calculus is a prerequisite for the class, there are a number of good examples and homework problems that the professor might use that would indeed require calculus.

Is discrete math above calculus? ›

Objects studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry.

Do you need to be good in math to be good in discrete math? ›

What math do I need to learn before discrete mathematics? Students with a solid understanding of algebra, geometry, and precalculus will do very well in discrete math.

Is IQ a discrete random variable? ›

IQ - discrete.

IQ scores are always integers - 100, 110, 180, etc.

Is blood type continuous or discrete? ›

Blood type is not a discrete random variable because it is categorical. Continuous random variables have numeric values that can be any number in an interval. For example, the (exact) weight of a person is a continuous random variable. Foot length is also a continuous random variable.

### Is eye color discrete or continuous? ›

Discrete data is organised in separate categories and is often presented in a bar chart. Examples of types of discrete data include => eye colour, hair colour, number of people in a shop, favourite chocolate bar.

What are 4 examples of continuous variables? ›

Therefore, at a macroscopic level, the mass, temperature, energy, speed, length, and so on are all examples of continuous variables.

What is an example of a continuous random variable in real life? ›

For example, the height of students in a class, the amount of ice tea in a glass, the change in temperature throughout a day, and the number of hours a person works in a week all contain a range of values in an interval, thus continuous random variables.

Is height discrete or continuous? ›

For example, the height of a student is a continuous variable because a student may be 1.6321748755... metres tall.

Why is blood pressure a continuous variable? ›

Arterial blood pressure (BP) is a continuous variable, with a physiology characterized by significant variability stemming from the complex interaction among haemodynamic factors, neuronal reflexes, as well as hormonal, behavioural, and environmental stimuli.

Is gender discrete or continuous? ›

Variable Reference Table : Few Examples
VariableVariable TypeVariable Scale
GenderDiscreteCategorical
Gender as Binary 1/0 CodingDiscreteCategorical
True/FalseDiscreteCategorical
Phone NumberDiscreteNominal
15 more rows
Oct 7, 2016

Is weight continuous or discrete? ›

Height, weight, temperature and length are all examples of continuous data. Some continuous data will change over time; the weight of a baby in its first year or the temperature in a room throughout the day.

Is normal distribution discrete or continuous? ›

The normal (Gaussian) and Lorentzian distributions are good examples of continuous distributions—the random variable can take on any value. Examples of discrete distributions include the Binomial, the Hypergeometric, and the Poisson.

Can continuous random variable assume any value between two intervals? ›

The correct answer is A. A continuous random variable may assume any value in an interval or collection of intervals. If X is a continuous random variable then it can take any value within an interval. Some examples of continuous random variables are height, weight, income, etc.

Can a continuous variables take any value between two numbers? ›

A continuous variable is defined as a variable which can take an uncountable set of values or infinite set of values. For instance, if a variable over a non-empty range of the real numbers is continuous, then it can take on any value in that range.

### Is Speed a continuous or discrete? ›

Yes, speed is a common continuous variable, and the value is chosen by a random process. We know it is continuous because there is always another possible value between any two speed values. It would not be possible to count all of the possible speeds that could be.

Can a continuous random variable be negative? ›

Whenever the value of a random variable is measured rather than counted, a continuous random variable is defined. The values of the random variables in these examples can be any of an infinite number of values within a defined interval [a,b]. These random variables could be any number from minus ∞ to plus b.

Do all continuous random variables have normal distributions? ›

No. There are many continuous probability distributions out of all the probability distributions.

How do you test the relationship between two continuous variables? ›

In statistical terms, correlation is a method of assessing a possible two-way linear association between two continuous variables. Correlation is measured by a statistic called the correlation coefficient, which represents the strength of the putative linear association between the variables in question.

How do we find probabilities for continuous random variables? ›

Similarly, the probability density function of a continuous random variable can be obtained by differentiating the cumulative distribution. The c.d.f. can be used to find out the probability of a random variable being between two values: P(s ≤ X ≤ t) = the probability that X is between s and t.

What is the sum of all the probabilities in continuous random variable? ›

Because all possible values of the random variable are included in the probability distribution, the sum of the probabilities is 1.

What are the two requirements for a discrete probability distribution? ›

Definition: probability distribution

The probabilities in the probability distribution of a random variable X must satisfy the following two conditions: Each probability P(x) must be between 0 and 1: 0≤P(x)≤1. The sum of all the possible probabilities is 1: ∑P(x)=1.

What are three examples of continuous variables? ›

Examples of continuous variables are body mass, height, blood pressure and cholesterol. A discrete quantitative variable is one that can only take specific numeric values (rather than any value in an interval), but those numeric values have a clear quantitative interpretation.

What are the properties of a continuous random variable? ›

A continuous random variable has two main characteristics: the set of its possible values is uncountable; we compute the probability that its value will belong to a given interval by integrating a function called probability density function.

What are 5 examples of discrete data? ›

Examples of discrete data
• The number of product reviews.
• The number of tickets sold in a day.
• The number of students in your class.
• The number of employees in a company.
• The number of computers in each department.
• The number of customers who bought different items.
• The number of items you buy at the grocery store each week.
Apr 12, 2023

### Is population discrete or continuous? ›

Population counts are typically referred to as discrete or quantitative data.

What are 3 examples of discrete data? ›

Some examples of discrete data would include:
• The number of employees in your department.
• The number of new customers you signed on last quarter.
• The number of products currently held in inventory.

Is gender a continuous variable? ›

Gender is a nominal discrete (and, in this study, binary) variable while upper-body strength (through various measurements in pounds) is a ratio continuous variable.

What is the difference of discrete variables and continuous variables? ›

Discrete and continuous variables are two types of quantitative variables: Discrete variables represent counts (e.g. the number of objects in a collection). Continuous variables represent measurable amounts (e.g. water volume or weight).

What is the difference between a discrete random variable and a continuous random variable quizlet? ›

What is the difference between a discrete random variable and a continuous random variable? A discrete random variable has a countable number of possible values. A continuous random variable has an infinite number of possible values, all the vlaues in an interval.

What is the main difference between discrete and continuous? ›

The primary difference, though, between discrete and continuous data is that discrete data is a finite value that can be counted whereas continuous data has an infinite number of possible values that can be measured.

What is the difference between continuous and discrete continuous? ›

The key differences are: Discrete data is the type of data that has clear spaces between values. Continuous data is data that falls in a constant sequence. Discrete data is countable while continuous — measurable.

What are examples of discrete and continuous variables? ›

Difference between Discrete and Continuous Variable
Discrete VariableContinuous Variable
Examples: Number of planets around the Sun Number of students in a classExamples: Number of stars in the space Height or weight of the students in a particular class
3 more rows
Oct 12, 2020

What is the main difference between discrete and continuous probability distribution? ›

A discrete distribution is one in which the data can only take on certain values, for example integers. A continuous distribution is one in which data can take on any value within a specified range (which may be infinite).

What is the difference between continuous and discrete types of data give one example of each type? ›

Values: Discrete data represents exact figures you can count, such as the numbers of students in a class. In contrast, continuous data often includes measurable values representing a range of information, such as the extent of the difference between the shortest and tallest student in a class.

### What is the difference between a discrete variable and a continuous variable choose the correct answer below quizlet? ›

A discrete variable has possible values that are separate​ numbers, while a continuous variable has possible values that form an interval.

What are the 3 differences of discrete and continuous? ›

Key Differences Between Discrete and Continuous Data

Discrete data is the type of data that has clear spaces between values. Continuous data is data that falls in a continuous sequence. Discrete data is countable while continuous data is measurable. Discrete data contains distinct or separate values.

What are the similarities and differences between continuous and discrete variables? ›

Discrete variables are the variables, wherein the values can be obtained by counting. On the other hand, Continuous variables are the random variables that measure something. Discrete variable assumes independent values whereas continuous variable assumes any value in a given range or continuum.

What is an example of a discrete variable? ›

As opposed to a continuous variable, a discrete variable can assume only a finite number of real values within a given interval. An example of a discrete variable would be the score given by a judge to a gymnast in competition: the range is 0 to 10 and the score is always given to one decimal (e.g. a score of 8.5).

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