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## Statistics and probability

### Course: Statistics and probability > Unit 9

Lesson 1: Discrete random variables- Random variables
- Discrete and continuous random variables
- Constructing a probability distribution for random variable
- Constructing probability distributions
- Probability models example: frozen yogurt
- Probability models
- Valid discrete probability distribution examples
- Probability with discrete random variable example
- Probability with discrete random variables
- Mean (expected value) of a discrete random variable
- Expected value
- Mean (expected value) of a discrete random variable
- Expected value (basic)
- Variance and standard deviation of a discrete random variable
- Standard deviation of a discrete random variable

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# Discrete and continuous random variables

Discrete random variables can only take on a finite number of values. 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. For example, the mass of an animal would be a continuous random variable, as it could theoretically be any non-negative number. Created by Sal Khan.

## Want to join the conversation?

- Can there really be any value for time? Isn't there a smallest unit of time? And if there isn't shouldn't there be?

If there is such a unit, then the variable cannot be continuous between one smallest unit and the next.(71 votes)- I think the smallest value of time is currently thought to be Planck time (time required for light to travel 1 planck length)

http://en.wikipedia.org/wiki/Planck_time

so I'm not quite sure whether time is actually discrete or continuous(80 votes)

- About the New Orleans Zoo example...

I think when we say "discrete" it means "countable" or "enumerable". In zoo example, although the exact mass would vary from almost 0 to say 5000kg (if that 5T elephant is indeed the biggest animal found in the zoo), we can count every individual animal at any moment. In other words, then total number of animal is somehow fixed at one moment. Therefore, I might say your zoo example is also an example of "discrete random variable".

How do you think?(33 votes)- What "discrete" really means is that a measure is separable. For instance, how many elephants does a zoo have? A zoo might have six elephants or seven elephants, but it can't have something between those two. Also, all zoos that have seven elephants definitely have the same number of elephants. Those two features make the number of elephants owned a discrete measure.

By contrast, the weight of an elephant is a continuous measure. If we say that one elephant weighs 945kg and another weighs 946kg, there are any range of possible weights between those two things. In fact, it is impossible for an elephant to weigh 945kg; we really mean that it is close enough to 945kg for our satisfaction. Unlike the count of elephants, a herd of elephants each of whom weighs 945kg each actually weighs something different. These are the clues that weight is a continuous measure and not discrete. Clear?(41 votes)

- Why is the word "random" in front of variable here. What's the difference between a discrete variable and a discrete random variable?(0 votes)
- It might be useful to watch the video previous to this, "Random Variables". He explains quite well how variables and random variables differ.(25 votes)

- so the distinction between discreet and continues random variables is determined by whether or not the possible outcomes are infinitely divisible into more possible outcomes?(13 votes)
- Essentially, yes. It's a nice way of thinking about it.(2 votes)

- I'm struggling to find a rigorous definition of discrete vs continuous. There are a lot of examples of discrete variables which produce integers as data but this doesn't seem to be the definition and I can think of many examples which do not adhere to this. For example:

X = "The number of blonde people divided by the total number of people in the elevator in my office whenever the door opens on the 5th floor." This random variable could yield rational numbers like 7/9 that are not integers, but which seem to be discrete. Is this discrete?

or X = "The length of a hypotenuse c, of a right angle triangle, where the lengths of sides a and b are determined by the roll of two 6 sided dice." Is this a discrete variable? I think it is but I'm not sure. There are only 36 possible lengths of hypotenuse c, but there can be irrational numbers like the square root of 2.

Or what if we defined that last example more mathematically and didn't constrain the lengths of sides a and b to 36 combinations?

Like, X = "the square root of a^2 + b^2, where a and b are any two random integers."

Now there are an infinite number of possible outcomes. Is this variable discrete? I still think it is, but I'm not sure because the definition of discrete and continuous seems a bit vague.

Tell me if you think this is an okay definition for a continuous variable : "A variable that can have an infinite number of possible values within ANY selected range." So in the zoo example that Sal gives at2:50, if you look for any value, even within the range of 1 elephant, you can get an infinite number of possible values. And then discrete is anything that doesn't meet this strict definition of continuous. Does this sound right?(6 votes)- Good points. Your definition is very close, but to spare yourself a few technicalities (the range of 0 elephants, for example), I would use the definition:

"A discrete variable is one that can take on*finitely*many, or*countably infinitely*many values", whereas a continuous random variable is one that is not discrete, i.e. "can take on*uncountably infinitely*many values", such as a spectrum of real numbers.

Your Pythagorean X is a good example. Although there are infinitely many possible values, they are still countable (because the combinations of a and b are countable), so X is indeed discrete.(9 votes)

- Would the winning time for a horse running in the Kentucky Derby (measured at 121 seconds or 121.25 seconds, for example) be classified as a discrete or continuous variable ?(3 votes)
- Based on the video, it depends on how time is recorded.

For a digital clock with a precision of 1/10,000 of a second, the number of discrete outcomes between 0 and 1 second (exclusive) is 9,999 ( 1/10,000 ... 9,999/10,000 ). Digital clocks and mechanical clocks with ratchets (the ones that tick) all produce discrete positions and the random variable would be discrete.

For a mechanical clock with a sweeping hand--no ratchet (doesn't tick)--the number of outcomes between 0 and 1 second would be infinite. Position the hand between 0 and 1. Now move the hand toward 0, then toward 1, now toward 0, and so on. You could put the hand in a new position each time and it would never repeat any previous position. The random variable is continuous.(10 votes)

- the exact time of the running time in the 2016 Olympics even in the hundredths is still continuous because it is still very hard to get to count a hundredth of a minute. If we do this couldn't we even count thousandths. That was my only problem but still great video and is helping me a lot for my slope test. Way better than my textbook, but still that was kind of confusing.(3 votes)
- I think the point being made is that the exact time it takes to do something is a continuous, while any sort of measurement and recording of the time, no matter how precise it may seem, is discrete since we have to cut off that precision at some point when measuring.(4 votes)

- and conversely, sometimes a discrete variable is actually treated continuously, such as population growth, even though strictly you can't have divisions of people , (what is a 13.43 people?) THe reason why is because we can use the tools of calculus to analyze population growth, and also because the sample space is so large (in the millions or billions), that it is relatively continuous. of course if your population is tiny you might want to use a discrete variable.(4 votes)
- whats the diffrence between the graph of a set of discrete data and the graph set of continouse data ?(3 votes)
- i think there is no graph (a line, or curve) for a set of discrete data. A graph presents a set of continuous data. Because a line, no matter how small it is, it must have the beginning point and the end point. The difference between 2 points is a collection of infinite points. Actually, a point itself is an infinite number. There is no point. There is nothing to be exact. so... we just make all the things up to define the world with less difficulties.

That's my opinion, i am not sure whether it helps your answer completely solved.(1 vote)

- At about10:20Sal explains that variable X is continuos because you can't count the exact times that it could be, but I thought time wasn't infinitely divisible. Am I wrong or what's going on here?(3 votes)
- I've been studying math now for over a month with the assistance of Khan academy. I begun from basic arithmetic and now I'm here. No problem so far and math has never before been this easy for me. Thank you so much for the work you do, the lessons are really educative.(1 vote)

## Video transcript

We already know a little
bit about random variables. 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 about. 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 think about another one. 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 think about another one. Let's think about-- 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.