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# Discrete and continuous random variables

## Video transcript

we already know a little bit about random variables what we're going to see in this video is that they're random variables come in two varieties you have discrete random variables and you have continuous random variables continuous 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 the meaning of the word discrete in the English language distinct distinct or separate separate values while continuous and I guess they're just another definition for the word de screening of language should be polite or not obnoxious or kind of subtle that is not what we're talking about rent we are not talking about random variables that are polite we're talking about ones that are just that 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 any value in an interval in an interval 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 it is equal to well this is one that we covered in the last video it's one if if my fair coin is heads it's zero if my fair coin is tails so is this a discrete or a continuous random variable well this Vallot this random variable right over here can take on distinct values it can take on either a 1 or it could take on a 0 another way to think about 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 this is clearly a discrete this is a discrete random variable discrete random let's think about another one let's define random variable Y as equal to the mass the mass of a random random animal animal selected at the selected at the New Orleans zoo New Orleans zoo where I grew up the ottoman zoo at the New Orleans at the New Orleans zoo the Audubon zoo why 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 the end 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 a part of that object right at that moment who knows the neutrons the protons the exact number of molecules in that in that object or part of that animal exactly that moment so that mass for example at the zoo it might take on a value anywhere between well maybe close to zero close to zero there's no animal that has zero 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 even a maybe if you consider even a bacterium an animal I believe bacterium is a 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 say 5,000 5,000 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 maybe something fun for you to look at but any animal could have a mass anywhere in between here there it does not take on discrete values you could have a animal that is exactly that is exactly 123 point seven five nine two one kilograms and even there that's actually might not be the exact mass you might have to get even more precise one zero seven three two and I get I think you get the picture there it could there's a even though this is the way I've defined now a finite interval you can take on any value in between here there are not discrete values so this one is clearly this is one here is continually a continuous random variable continuous 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 the year that a random student in the class was born random student in a 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 it could be 1992 or it could be 1985 or it could be 2000 and 2001 there are discrete values that this random variable can actually 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 is in the case right here a discrete a discrete random variable and let me make it clear this one over here is also a discrete random variable most of the time that you're dealing with the discrete random variable you are 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 a potential number of values that it could take on as long as the values are countable as long as you can literally say okay this is the first value can take on the second the third and you might be counting forever but as long as you can literally it could be even an infinite list but if you could list the values that it could take on then you're dealing with the 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 could you know you might attempt to and it's 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 okay maybe we can you 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 zero point zero 1 1 zero point zero one two and even between those there's an infinite number of values you 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 of ants number of ants born tomorrow born tomorrow in the universe in the universe now you're probably arguing that there aren't ants on other planets or maybe they're ant light creatures but they're they're not going to be ants as we define them but how do we know so a 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 in 1 we can actually count the values those values are discrete so once again this right over here is a discrete 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 well let's say I random variable X so we're not we're not using this definition anymore now I'm going to define random variable X to be the winning time let me write it this way the exact winning time exact time winning time for the men's men's 100-meter 100-meter dash at the 2016 Olympics 2016 Olympics so the exact time that it took for the winner who's probably going to be Hussain 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 dash of 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 continuous random variable well the way I've defined it and this one's a little bit tricky because you might say it's countable you might say well it could either be it could be 956 nine point five six seconds or nine point five seven seconds or nine point five eight seconds and you might be tempted to believe that because when you watch the hundred meter dash at the Olympics they measure it to the nearest hundredths they round to the nearest hundredths that's how precise their timing is but I'm talking about the exact 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 and you know it might be anywhere between five seconds five seconds and maybe 12 12 seconds it could be anywhere in between there it might not be nine point five seven that might be what the clock says but in reality the exact winning time could be nine point five seven one or it could be nine point five seven two three five nine 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 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 rounded to the nearest the nearest hundred nearest hundredths hundredths 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 continuous random variable well now we can actually count the actual values that this random variable can take on it might be nine point five six it could be nine point five seven it could be nine point five eight we can actually list them so in this case when we rounded to the nearest hundredths we can actually list the values we are now just dealing with a discrete discrete random variable 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