List of stochastic processes topics

Transcription

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 discreet 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 discreet 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 discreet 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.