Learn about Probability Density Functions Video

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Learn about Probability Density Functions

In the last video I introduce to you to the notion of a probability or really we start with the random variable and then we move on to the two types of random variables. You have discrete that took on a finite number of values and this. Well, I almost going to say that they tend to be integers but they don’t always have to be integers. You have discrete random and so finite meaning that you can’t have an infinite number of values for discrete random variable. Then we have the continuous which can take on continuous, an infinite number and the example I give for continuous is let’s say random variable x and people do tend to use, let me change it a little just so you can see it can be something other an x. Let’s say I have the random variable capital Y. They do tend to be capital letters is equal to the exact amount of rain tomorrow and I say rain because I’m in Northern California, it’s actually raining quite hard right now which where short right now. So, that’s a positive and we’ve been having a drought. So, it’s a good thing but the exact amount of rain tomorrow. Let’s say, I don’t know what the actual probability distribution function for this is but I’ll draw one and then we will interpret it just so you can kind of think about how can think about continuous random variables. So, let me draw its probability distribution and they called it’s the probability density function and we draw it like this. And let’s say that there is it looks something like this and now like that, alright. Then I don’t know what this height it is. The x axis here is the amount of rain. Well, this is zero inches. This is one inch. This is two inches. This is three inches, four inches, and then this is some height. Let’s say it picks out here at, I don’t know, let’s say this is .5. So, the way you think about it, if you were to look at this and I were to ask you: what is the probability that y because that’s a random variable. Now that y is exactly equal to two inches, then y is exactly equal to two inches, right. What’s the probability with that happening? Well, based on how we thought about the probability distribution functions for the discrete random variable you’ve say, “Okay, let’s see two inches, that’s the case we care about right now.” Let me go up here and say, “Okay. It looks like it about .5” and you say, “Well, I don’t know is that a .5 chance.” And I would say, “No, it is not a .5 chance and before we even think about that how we would interpret it visually, let’s just think about it logically. What is the probability that tomorrow we have exactly two inches of rain, not 2.01 inches of rain and not 1.99 inches of rain, not 1.9999 inches of rain, not 2.00001 inches of rain, exactly 2 inches of rain. You know I mean it like you know; there is not a single extra atom of water molecule above the two inch mark and not single water molecular below the two inch mark. It’s essentially zero, right? It might not be obvious to you because you probably heard, “Oh, you know we have two inch rain last night.” But think about the exactly two inches, right. Normally if it’s like 2.01, people will say that is two but we’re saying no, that this does not count. It can’t be two as we want the exactly two, 1.99 does not count. Normally, I mean our measurements we don’t even have tools that can tell us whether it is exactly two inches, right. No matter, no ruler you can even say is exactly two inches long. At some point you know, just the way we manufacture things there is going to be an extra atom on it here or there. So, the odd of actually anything being exactly a certain measurement to the exact infinite decimal point is actually zero. The way you would think about a continuous random variable you could say what is the probability that you know, y is almost two. So, if we said that the absolute value of y-2 is less than some tolerance, it’s less than .1, right. And if that doesn’t make sense to you, this is essentially just saying that what is the probability that y is greater than 1.9 and less than 2.1. These two statements are equivalent. I will let you to think about a little bit but now this starts to make a little bit sense. Now we have an interval here. So, we want all the y’s between 1.9 and 2.1. So, we are now talking about this whole area. An area is a key so if you want to know the probability of this occurring, you actually want the area under this curve from this point to this point and for those of you have studied your calculus that would essentially be the definite integral of this probability density function from this point to this point. So, let’s say if this line was defined by I call it f(x), I call it p(x) or something. The probability of this happening would be equal to the integral for those of you who studied calculus from 1.9 to 2.1 of f(x) and assuming you know this is the x axis, right. So, it’s a very kind of important thing to realize because when a random variable can take an infinite number of values or it can take any value between the intervals to get an exact value. To get exactly 1.99 and the probability is actually zero. It’s like asking you what is the area under a curve on just this line. Well or even more specific because it’s like asking you what is the area of a line and the area of a line if you were just to draw a line and you say, well area is height times base. Well, the height has some dimension but the base, what’s the width of a line, right. As far as, you know, the way that we have to find a line. A line has no width and therefore no area and it should make intuitive sense that the probability of a very super exact thing happening is pretty much zero. Then you really have to say, okay what is the probability that were get close to two? Then you can define an area and if you said, What is the probability that we get in some place one and three inches of rain and that off course the probability is much higher with all of this kind of stuff, right. You could also say you know, what is a probability we have less than .1 inches of rain? Then you would go here and you would calculate this was .1 and you would calculate this area and you could say what is the probability that we have more than four inches of rain tomorrow? Then you would start here and you would calculate the area on the curve all the way to infinity if the curve has area all it way into infinity. And hopefully that is not an infinite number, right. Then your probability won’t make any sense but hopefully if you take this sum, it comes to some number and it will, there only have 10% chance that you have more than four inches tomorrow and all of this should kind of immediately lead to one light bulb in you head is that the probability of all of the events that might occur can’t be more than a 100% right. All of the events combined and can’t you know, that there is a probability of one that one of this events will occur. So, essentially the whole area under this curve has to be equal to one. So, if we took the integral of f(x) from zero to infinity, this thing at least as I have drawn it dx should be equal to one for those of you who have studied calculus. For those of you who having an integral, it just the area under the curve and you can watch the calculus videos if you want to learn a little bit more about how to do them. And this also applies to the discrete probability distributions. Let me draw one, the sum of all of the probabilities have to be equal to one and that example with the dice or let’s say the, since it’s faster to draw the coin. The two probabilities have to be equal to one. So, if this is (1, 0) where x is equal to one if we’re heads or zero if we’re tails. Each of this has to be .5 or they don’t have to be .5 but if one was .6, then you only have to be .4. They have to add it to one. If one of this was, that you can’t have a 60% probability of getting a heads and then a 60% probability getting a tails as well because then you would have essentially a 120% probability of either of the outcomes happening which makes no sense at all. So, it’s important to realize that the probability distribution function in this case for discrete or random variable, they all have to add to one is .5 plus .5. And in this case, the area under the probability density function also has to be equal to one. Anyway I’m all out of time for now. In the next video, I’ll introduce you to the idea of an expected value. See you soon.

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In the last video I introduce to you to the notion of a probability or really we start with the random variable and then we move on to the two types of random variables. You have discrete that took on a finite number of values and this. Well, I almost going to say that they tend to be integers but they don’t always have to be integers. You have discrete random and so finite meaning that you can’t have an infinite number of values for discrete random variable.

Then we have the continuous which can take on continuous, an infinite number and the example I give for continuous is let’s say random variable x and people do tend to use, let me change it a little just so you can see it can be something other an x. Let’s say I have the random variable capital Y. They do tend to be capital letters is equal to the exact amount of rain tomorrow and I say rain because I’m in Northern California, it’s actually raining quite hard right now which where short right now.

So, that’s a positive and we’ve been having a drought. So, it’s a good thing but the exact amount of rain tomorrow. Let’s say, I don’t know what the actual probability distribution function for this is but I’ll draw one and then we will interpret it just so you can kind of think about how can think about continuous random variables. So, let me draw its probability distribution and they called it’s the probability density function and we draw it like this. And let’s say that there is it looks something like this and now like that, alright.

Then I don’t know what this height it is. The x axis here is the amount of rain. Well, this is zero inches. This is one inch. This is two inches. This is three inches, four inches, and then this is some height. Let’s say it picks out here at, I don’t know, let’s say this is .5. So, the way you think about it, if you were to look at this and I were to ask you: what is the probability that y because that’s a random variable. Now that y is exactly equal to two inches, then y is exactly equal to two inches, right. What’s the probability with that happening?

Well, based on how we thought about the probability distribution functions for the discrete random variable you’ve say, “Okay, let’s see two inches, that’s the case we care about right now.” Let me go up here and say, “Okay. It looks like it about .5” and you say, “Well, I don’t know is that a .5 chance.” And I would say, “No, it is not a .5 chance and before we even think about that how we would interpret it visually, let’s just think about it logically. What is the probability that tomorrow we have exactly two inches of rain, not 2.01 inches of rain and not 1.99 inches of rain, not 1.9999 inches of rain, not 2.00001 inches of rain, exactly 2 inches of rain.

You know I mean it like you know; there is not a single extra atom of water molecule above the two inch mark and not single water molecular below the two inch mark. It’s essentially zero, right? It might not be obvious to you because you probably heard, “Oh, you know we have two inch rain last night.” But think about the exactly two inches, right. Normally if it’s like 2.01, people will say that is two but we’re saying no, that this does not count. It can’t be two as we want the exactly two, 1.99 does not count.

Normally, I mean our measurements we don’t even have tools that can tell us whether it is exactly two inches, right. No matter, no ruler you can even say is exactly two inches long. At some point you know, just the way we manufacture things there is going to be an extra atom on it here or there. So, the odd of actually anything being exactly a certain measurement to the exact infinite decimal point is actually zero.

The way you would think about a continuous random variable you could say what is the probability that you know, y is almost two. So, if we said that the absolute value of y-2 is less than some tolerance, it’s less than .1, right. And if that doesn’t make sense to you, this is essentially just saying that what is the probability that y is greater than 1.9 and less than 2.1. These two statements are equivalent. I will let you to think about a little bit but now this starts to make a little bit sense. Now we have an interval here.

So, we want all the y’s between 1.9 and 2.1. So, we are now talking about this whole area. An area is a key so if you want to know the probability of this occurring, you actually want the area under this curve from this point to this point and for those of you have studied your calculus that would essentially be the definite integral of this probability density function from this point to this point.

So, let’s say if this line was defined by I call it f(x), I call it p(x) or something. The probability of this happening would be equal to the integral for those of you who studied calculus from 1.9 to 2.1 of f(x) and assuming you know this is the x axis, right. So, it’s a very kind of important thing to realize because when a random variable can take an infinite number of values or it can take any value between the intervals to get an exact value. To get exactly 1.99 and the probability is actually zero. It’s like asking you what is the area under a curve on just this line.

Well or even more specific because it’s like asking you what is the area of a line and the area of a line if you were just to draw a line and you say, well area is height times base. Well, the height has some dimension but the base, what’s the width of a line, right. As far as, you know, the way that we have to find a line. A line has no width and therefore no area and it should make intuitive sense that the probability of a very super exact thing happening is pretty much zero. Then you really have to say, okay what is the probability that were get close to two?

Then you can define an area and if you said, What is the probability that we get in some place one and three inches of rain and that off course the probability is much higher with all of this kind of stuff, right. You could also say you know, what is a probability we have less than .1 inches of rain? Then you would go here and you would calculate this was .1 and you would calculate this area and you could say what is the probability that we have more than four inches of rain tomorrow?

Then you would start here and you would calculate the area on the curve all the way to infinity if the curve has area all it way into infinity. And hopefully that is not an infinite number, right. Then your probability won’t make any sense but hopefully if you take this sum, it comes to some number and it will, there only have 10% chance that you have more than four inches tomorrow and all of this should kind of immediately lead to one light bulb in you head is that the probability of all of the events that might occur can’t be more than a 100% right. All of the events combined and can’t you know, that there is a probability of one that one of this events will occur.

So, essentially the whole area under this curve has to be equal to one. So, if we took the integral of f(x) from zero to infinity, this thing at least as I have drawn it dx should be equal to one for those of you who have studied calculus. For those of you who having an integral, it just the area under the curve and you can watch the calculus videos if you want to learn a little bit more about how to do them.

And this also applies to the discrete probability distributions. Let me draw one, the sum of all of the probabilities have to be equal to one and that example with the dice or let’s say the, since it’s faster to draw the coin. The two probabilities have to be equal to one. So, if this is (1, 0) where x is equal to one if we’re heads or zero if we’re tails. Each of this has to be .5 or they don’t have to be .5 but if one was .6, then you only have to be .4.

They have to add it to one. If one of this was, that you can’t have a 60% probability of getting a heads and then a 60% probability getting a tails as well because then you would have essentially a 120% probability of either of the outcomes happening which makes no sense at all. So, it’s important to realize that the probability distribution function in this case for discrete or random variable, they all have to add to one is .5 plus .5. And in this case, the area under the probability density function also has to be equal to one. Anyway I’m all out of time for now. In the next video, I’ll introduce you to the idea of an expected value. See you soon.

Transcription by: Scribe4you Transcription Services