Random Variables Explained: Discrete vs Continuous | Probability

[00:00:00]I'll now introduce you to the concept of a random variable.

[00:00:06]And for me this is something that I always had a lot of trouble getting my head around, and that's really because it's a byproduct of what it's called.

[00:00:14]It's called a variable and we're used to variables as kind of an unknown in the equation.

[00:00:18]If I write x plus 3 is equal to 7, the variable was x.

[00:00:24]Maybe you could solve for it or maybe you could have an equation y is equal to 3x minus 2.

[00:00:30]And then, here y and x are both variables.

[00:00:33]If you input 1x you could solve for the other variable, y.

[00:00:36]And you can change them.

[00:00:38]And variables were kind of things that could change and that you could solve for.

[00:00:41]And they could take on particular values.

[00:00:44]A random variable is kind of the same thing in that it can take on multiple values, but it's not something that you really ever solve for.

[00:00:51]So just so you get used to the notation, a random variable is usually a capital letter.

[00:00:56]Usually a Capital X, Y, or Z.

[00:00:58]Usually a capital X.

[00:00:59]And where it really defers from a traditional variable is that it can take on a bunch of different values, like a traditional variable but you never solve for it.

[00:01:06]And really, it's a little misleading to call it a variable at all.

[00:01:09]It's really a function.

[00:01:11]And it's a function that maps you from the world of random processes to an actual number.

[00:01:17]So let's say-- I don't know-- I wanted to somehow quantify a random process.

[00:01:23]Is it going to rain or not tomorrow?

[00:01:25]So let's see-- rain tomorrow.

[00:01:31]No rain.

[00:01:34]So you could observe that.

[00:01:36]You could wait until tomorrow and see if it rains or not, but then, how do you quantify it?

[00:01:39]Well, we can define a random variable that will quantify it.

[00:01:42]We can say this random variable is going to be equal to 1 if it rains tomorrow and it equals 0 if it doesn't rain tomorrow.

[00:01:50]We didn't have to assign 1 and 0; those tend to be a little bit more useful.

[00:01:53]They make sense.

[00:01:54]But we could have assigned this as-- I don't know-- we could have said that this is 21 and that this is 100.

[00:01:59]It's however you define it.

[00:02:00]So it's important to keep this distinction in mind, that a random variable-- it isn't a variable in the traditional sense of the world.

[00:02:07]It's more of a function that maps us from the world of a random process to a number.

[00:02:15]And then, this number is going to be random.

[00:02:18]Because we don't know if it's going to rain or not tomorrow, maybe we have some sense of the probabilities.

[00:02:23]So this variable can take on either value.

[00:02:27]And that leads us-- let me draw a couple of more random variable definitions just so you get the intuition that these really are functions.

[00:02:37]So we could define a random variable x as equal to-- well, actually, let me-- It could just be a very obvious numeric mapping.

[00:02:45]It could just be the number facing up when I roll a fair dice.

[00:02:58]That's a random variable.

[00:03:00]I could've also said just the number most facing east when I rolled a fair die, although that's a little less useful.

[00:03:06]You could also define a random variable, x is equal to-- you could say it's equal to 1 if heads, and I could say 0 if tails.

[00:03:21]Or I could have done it the other way around.

[00:03:22]Once again, these heads and tails are outcomes of an experiment.

[00:03:27]When I flip a coin that's a random process, each flip is an experiment and then the random variable is just quantifying that experiment.

[00:03:37]So I know it sounds a little technical with the terminology, but what we're really doing is fairly something simple.

[00:03:41]We're just assigning a 1 if we get a heads, a 0 if we get a tails.

[00:03:44]And the random variable is just that function mapping.

[00:03:49]Now I'm going into all of this random variable business, which I didn't do before when we did probability because it does start to become a little bit useful notationally.

[00:03:58]Because we'll start to talk about things like probability distributions and expected values, and it really is useful to quantify things as a random variable.

[00:04:06]And with that said, let's talk about probability distributions and expected values.

[00:04:12]So the first thing, probability-- actually, let me define something else.

[00:04:17]There's two types of random variables.

[00:04:20]You can have discrete random variables, or continuous.

[00:04:28]So there's two types.

[00:04:29]Discrete would be like, really all of the definitions that we had up until now.

[00:04:35]The flip of a coin, you're either going to get a 1 or a 0.

[00:04:38]There's two discrete cases.

[00:04:40]When I roll a die there's 1 of 6 different outcomes I can have.

[00:04:44]Whether it rains tomorrow, there's 1 of 2 outcomes: yes or no.

[00:04:49]So in all of these situations you could almost say you could have accountable number of outcomes.

[00:04:53]A continuous random variable could take on an infinite number of outcomes.

[00:04:58]So a continuous random variable would be the exact-- it could be x is equal to the exact amount of rain in inches tomorrow.

[00:05:13]Now why is this continuous?

[00:05:16]Well, if you think about it that could take on any of an infinite set of values.

[00:05:21]You might have 1 inch of rain tomorrow or you might have 1.1 inches of rain.

[00:05:25]You might have 1.111 inches of rain.

[00:05:28]You might have 2.1111 inches of rain.

[00:05:31]As you could see, you can come up with an infinite number of combinations of the amount of rain you have.

[00:05:37]To get exact-- well, I'll go into more of this when I talk about probability density functions.

[00:05:41]But in general, even though you might say, oh, well, I can't imagine having an infinite amount of rain because we're not going to have 100 gallons of rain tomorrow.

[00:05:49]But if you think about it, you could take on any value between 1 and 2 inches of rain.

[00:05:53]There's actually an infinite number of values between 0 and 1.

[00:05:59]For any value you could find a number that's a little-- just to think about it.

[00:06:04]Between 0 and 1, 1/2 is in between those.

[00:06:07]And you can always find a number that's halfway between those and then a number that's halfway between those.

[00:06:11]There's an infinite number of numbers between any two numbers.

[00:06:15]While, with the discrete random variables, it just took on a finite number of values.

[00:06:21]Those ones up here, it could only take on 1 or 0.

[00:06:23]It didn't take on 1.1.

[00:06:25]Or it couldn't take on infinity.

[00:06:27]And I make that difference because actually, how we look at it in terms of their probability distributions are a little different.

[00:06:35]But they're very related.

[00:06:37]So a discreet random variable like the ones that we had defined at the beginning of these videos, they have a probability distribution.

[00:06:44]Let's do it for-- let's say x is equal to the number facing up on a fair dice.

[00:07:02]So we already know that to get each outcome-- there's 6 outcomes.

[00:07:07]You either can get a 1, 2, 3, 4, 5, or 6 facing up.

[00:07:11]So let me draw the probability distribution.

[00:07:15]So here are all the outcomes.

[00:07:16]I'll draw that on the horizontal axis.

[00:07:19]So you get one outcome is 1, 2, 3, 4, 5, 6.

[00:07:30]And then, in the y-axis or in the vertical, you plot what is the probability of each of those outcomes occurring?

[00:07:38]What's the probability?

[00:07:40]I'll switch colors.

[00:07:42]So what's the probability of getting a 1?

[00:07:43]Well, it's 1/6.

[00:07:45]So if I draw 1/6 here.

[00:07:50]So the probability is of each of these occurring-- oh, I didn't want to do it like that-- are 1/6.

[00:07:56]It's essentially like a histogram or a bar chart.

[00:08:02]My intention is to draw them all the exact same height.

[00:08:08]So this is telling us that each of these there's a 1/6 chance of it occurring.

[00:08:14]If I were to draw a probability distribution for-- if I were to find the random variable x is equal to 1 if heads, 0 if tails.

[00:08:31]This has a very simple probability distribution.

[00:08:37]There's only 2 outcomes.

[00:08:39]You're either going to get a 1 or a 0.

[00:08:43]And then in the y-axis you'll have the probabilities.

[00:08:49]And each of these, if it's a fair dice, it's going to be a 1/2 probability.

[00:08:53]And you're going to draw a little box like that that tells you each of them are 1/2.

[00:08:59]Now let's say that you had another-- I don't know-- probably distribution function that looks something like this.

[00:09:04]Those are kind of very run of the mill examples, but what if I had something like this?

[00:09:10]Let's see.

[00:09:13]Where I had some dice-- let's say we had some weird weightings on it where it had an outcome of 1-- I had a 1/6 chance like that.

[00:09:28]Let's say for some reason a 2 could never happen.

[00:09:34]Maybe there is no 2.

[00:09:36]Let's say 3 has a 1/6 chance, a 4 has a 1/6 chance, a 5 has a 1/6 chance.

[00:09:44]And let's say there's two 6's.

[00:09:46]Like we erased the 2 and we put a 6 where the 2 is, so there's two 6's.

[00:09:50]So the 6 has a 2/6 chance.

[00:09:54]So the 6 would be a 2/6 chance.

[00:09:56]So now I think you can see where the probability distribution function starts to become a little bit more interesting.

[00:10:03]In the other ones it was kind of what we would call a uniform distribution.

[00:10:07]All of the outcomes are equally likely.

[00:10:09]That's a uniform distribution.

[00:10:10]Once again, this is a uniform distribution.

[00:10:13]This one, all of a sudden, is not a uniform distribution.

[00:10:16]1 is just as likely as 3, 4, 5.

[00:10:19]2 is impossible to happen, but 6 is twice as likely to happen as everything else-- is 2/6.

[00:10:25]If someone gives you a probability density function or if they give you a little chart like this, you can immediately say, what's the probability of different events occurring?

[00:10:34]So if you said that x is equal to the number on unfair dice that's described by this probability distribution function.

[00:10:48]Then if I were to ask you the probability that x is equal to 6, you'd say, oh, the probability x equals 6 is equal to 2/6 or 1/3.

[00:10:59]If I were to say what's the probability that x, the random variable x as given by this definition, is greater than 5, you would take-- well, let's say greater than or equal to 5.

[00:11:13]Because you want to include 5, and at least, I did.

[00:11:16]So it would include this situation and this situation.

[00:11:19]So it'd be 1/6 plus 2/6.

[00:11:25]And so you could say that's 3/6 or 1/2.

[00:11:30]So it gets pretty interesting once you don't have a completely uniform distribution.

[00:11:34]And I'm doing all of this because in the next couple of videos-- well, one, we'll move away from discreet and actually study continuous probability distributions, which are called probability density functions.

[00:11:44]And then we'll go to study kind of a lot of distributions that show up in nature.

[00:11:48]The uniform distribution is one of them.

[00:11:51]There's also the binomial distribution and the normal distribution, which is what some people call the Gaussian or the bell curve.

[00:11:58]Which is kind of a very common thing.

[00:11:59]But anyway, I'm all out of time and I'll see you in the next video.