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# Random variables

Basic idea and definitions of random variables. Created by Sal Khan.

Video transcript

What I want to discuss a
little bit in this video is the idea of a
random variable. And random variables at first
can be a little bit confusing because we will want to think
of them as traditional variables that you were first exposed
to in algebra class. And that's not quite what
random variables are. Random variables are
really ways to map outcomes of random processes to numbers. So if you have a random process,
like you're flipping a coin or you're rolling dice or you
are measuring the rain that might fall tomorrow,
so random process, you're really just mapping
outcomes of that to numbers. You are quantifying
the outcomes. So what's an example
of a random variable? Well, let's define
one right over here. So I'm going to define
random variable capital X. And they tend to be
denoted by capital letters. So random variable capital
X, I will define it as-- It is going
to be equal to 1 if my fair die rolls heads--
let me write it this way-- if heads. And it's going to be
equal to 0 if tails. I could have defined
this any way I wanted to. This is actually a
fairly typical way of defining a random variable,
especially for a coin flip. But I could have
defined this as 100. And I could have
defined this as 703. And this would still be a
legitimate random variable. It might not be as pure a
way of thinking about it as defining 1 as
heads and 0 as tails. But that would have
been a random variable. Notice we have taken this
random process, flipping a coin, and we've mapped the outcomes
of that random process. And we've quantified them. 1 if heads, 0 if tails. We can define another random
variable capital Y as equal to, let's say, the sum of
rolls of let's say 7 dice. And when we talk
about the sum, we're talking about the
sum of the 7-- let me write this-- the
sum of the upward face after rolling 7 dice. Once again, we are quantifying
an outcome for a random process where the random process
is rolling these 7 dice and seeing what
sides show up on top. And then we are taking those
and we're taking the sum and we are defining a
random variable in that way. So the natural
question you might ask is, why are we doing this? What's so useful about defining
random variables like this? It will become
more apparent as we get a little bit
deeper in probability. But the simple way
of thinking about it is as soon as you
quantify outcomes, you can start to do a little
bit more math on the outcomes. And you can start
to use a little bit more mathematical
notation on the outcome. So for example, if you
cared about the probability that the sum of the upward
faces after rolling seven dice-- if you cared
about the probability that that sum is less than
or equal to 30, the old way that you would have
to have written it is the probability
that the sum of-- and you would have to write
all of what I just wrote here-- is less than or equal to 30. You would have had to
write that big thing. And then you would try
to figure it out somehow if you had some information. But now we can just
write the probability that capital Y is less
than or equal to 30. It's a little bit
cleaner notation. And if someone else cares
about the probability that this sum of the upward
face after rolling seven dice-- if they say, hey, what's the
probability that that's even, instead of having to
write all that over, they can say, well, what's the
probability that Y is even? Now the one thing that
I do want to emphasize is how these are different
than traditional variables, traditional variables that
you see in your algebra class like x plus 5 is equal
to 6, usually denoted by lowercase variables. y is equal to x plus 7. These variables, you can
essentially assign values. You either can solve for
them-- so in this case, x is an unknown. You could subtract 5 from
both sides and solve for x. Say that x is going
to be equal to 1. In this case, you could say,
well, x is going to vary. We can assign a
value to x and see how y varies as a function of x. You can either
assign a variable, you can assign values to them. Or you can solve for them. You could say, hey x is
going to be 1 in this case. That's not going to be the
case with a random variable. A random variable can take on
many, many, many, many, many, many different values with
different probabilities. And it makes much
more sense to talk about the probability of
a random variable equaling a value, or the probability
that it is less than or greater than something,
or the probability that it has some property. And you see that in
either of these cases. In the next video, we'll
continue this discussion and we'll talk a little
bit about the types of random variables
you can have.