WEBVTT mathematics/probability/murray
00:00:00.000 --> 00:00:06.000
Hi there, these are the probability videos here on www.educator.com, my name is Will Murray.
00:00:06.000 --> 00:00:12.100
We are working through a chapter on Bivariate densities and Bivariate distributions
00:00:12.100 --> 00:00:15.700
which means we will have two variables, a Y1 and Y2.
00:00:15.700 --> 00:00:23.600
We have just been looking at some videos on marginal probabilities and also, on conditional probability.
00:00:23.600 --> 00:00:27.200
We are going to be using some of those ideas in this video.
00:00:27.200 --> 00:00:30.800
Today's video is on independent random variables.
00:00:30.800 --> 00:00:35.800
I will be using the notion of marginal density function.
00:00:35.800 --> 00:00:40.700
If you do not remember anything about marginal probability or marginal density functions,
00:00:40.700 --> 00:00:47.000
what you might want to do is just go back and just briefly review the idea of marginal density functions.
00:00:47.000 --> 00:00:50.100
Because, we are going to use that, we will use the definition of those,
00:00:50.100 --> 00:00:54.700
in this video today on independent random variables.
00:00:54.700 --> 00:00:56.700
Having said that, let us jump in.
00:00:56.700 --> 00:01:02.100
The intuition of independent random variables is sort of one thing.
00:01:02.100 --> 00:01:06.800
And then, there is a definition and then there is a theorem about independent random variables.
00:01:06.800 --> 00:01:09.300
There are three different ways to think about it.
00:01:09.300 --> 00:01:15.400
There is intuition, there is a definition, and then there is a theorem which is also very useful.
00:01:15.400 --> 00:01:22.100
I will spell out each one, I got a side on the intuition, and then the next side will be the formal definition,
00:01:22.100 --> 00:01:26.200
and then the next side will be the theorem that you sometimes want to use.
00:01:26.200 --> 00:01:32.500
The idea here is that, we have an experiment with two random variables, Y1 and Y2.
00:01:32.500 --> 00:01:41.300
Intuitively, independence means that if I tell you the value of Y2, I tell you the value of one of the variables,
00:01:41.300 --> 00:01:49.100
you really have no new information about the distribution of the other variable Y1.
00:01:49.100 --> 00:01:54.800
Maybe, you can make a prediction about Y1 and then I would tell you the value of Y2,
00:01:54.800 --> 00:01:57.500
and say do you want to change your prediction about Y1?
00:01:57.500 --> 00:02:02.000
If they are independent then no, you would not change your prediction of Y1
00:02:02.000 --> 00:02:10.100
because the new information about the value of Y2 does not tell you anything new about the value of Y1.
00:02:10.100 --> 00:02:14.300
That is the intuitive idea of independence.
00:02:14.300 --> 00:02:38.000
If we spell that out, in terms of equations, what we have here, F1 of Y1 is the marginal density function of Y1.
00:02:38.000 --> 00:02:56.000
What we have on the right here, F of Y1 condition on Y2 is the conditional density function of Y1.
00:02:56.000 --> 00:03:06.300
The idea is that on the left, this marginal density function of Y1, this is how you would describe Y1,
00:03:06.300 --> 00:03:10.400
if you have no information at all about Y2.
00:03:10.400 --> 00:03:18.900
If you knew nothing about Y2, this is how you think the density of Y1 behaves.
00:03:18.900 --> 00:03:21.600
On the right, we have the conditional density function.
00:03:21.600 --> 00:03:32.000
This is, if I told you a particular value of Y2, how would you describe the density of Y1 with that extra information.
00:03:32.000 --> 00:03:38.000
On the right is, how you would make predictions with the extra information about Y2.
00:03:38.000 --> 00:03:43.100
On the left is, how you would make predictions with no information at all about Y2.
00:03:43.100 --> 00:03:46.500
The idea of independence is that, those should be the same.
00:03:46.500 --> 00:03:52.800
The extra information about Y2, does not change what you know about Y1.
00:03:52.800 --> 00:03:57.700
That should be kind of intuitively why this formula makes sense.
00:03:57.700 --> 00:04:04.700
This is not actually the formula that we will use to check whether variables are independent.
00:04:04.700 --> 00:04:10.800
I’m going to give you a different definition on the next slide but I think this is sort of the more intuitive formula.
00:04:10.800 --> 00:04:16.700
After I gave you the new definition on the next slide, I will try to connect it back to this formula
00:04:16.700 --> 00:04:20.700
so that you see how the two ideas are related.
00:04:20.700 --> 00:04:25.700
I know that is a bit confusing to have two different ways of approaching something.
00:04:25.700 --> 00:04:29.200
I'm going to try to persuade you that they do both make sense, and that,
00:04:29.200 --> 00:04:34.000
you can get back and forth from one to the other.
00:04:34.000 --> 00:04:38.700
In the next slide, we are going to look at the formal definition of independence,
00:04:38.700 --> 00:04:45.200
which I think is a little less obvious but I will connect it back to this intuitive idea.
00:04:45.200 --> 00:04:51.700
The formal definition of independence for random variables, in terms of probability,
00:04:51.700 --> 00:05:00.700
is the probability of both of them taking a particular value is the same as, if you evaluate them separately
00:05:00.700 --> 00:05:07.000
and find the probability that each one of them takes that value separately and then multiply those probabilities.
00:05:07.000 --> 00:05:10.600
That is actually the discreet version of independence.
00:05:10.600 --> 00:05:19.900
The short version of that same formula is the probability of Y1, Y2 is the probability of Y1,
00:05:19.900 --> 00:05:25.700
the marginal probability of Y1 × the marginal probability of Y2.
00:05:25.700 --> 00:05:40.100
Let me stress here that this P1P2, those are marginal probabilities.
00:05:40.100 --> 00:05:46.200
On the left, we have the joint probability function.
00:05:46.200 --> 00:05:53.600
That was the discrete case and the continuous case is the analogue of that.
00:05:53.600 --> 00:06:05.100
On the left, we have the joint probability function.
00:06:05.100 --> 00:06:15.500
On the right, we have the two marginal probability functions, marginal density functions.
00:06:15.500 --> 00:06:25.800
The idea of independence is that the joint probability density function,
00:06:25.800 --> 00:06:30.800
maybe it would be better if I said joint density function instead of joint probability.
00:06:30.800 --> 00:06:33.500
Let me write that down.
00:06:33.500 --> 00:06:43.800
The idea of independence is that the joint density function is equal to the product of the two marginal density functions.
00:06:43.800 --> 00:06:52.400
Let me write that a little more clearly, densities.
00:06:52.400 --> 00:06:58.400
The joint density function factors into a product of the two marginal density functions.
00:06:58.400 --> 00:07:02.300
They sort of split apart and they are independent there.
00:07:02.300 --> 00:07:08.800
Let me try to connect this, this is the formal definition of independence.
00:07:08.800 --> 00:07:12.200
This is the one that we are going to use for most of the problems.
00:07:12.200 --> 00:07:21.400
Let me try to connect this up with the intuitive formula that I gave you back on the previous slide.
00:07:21.400 --> 00:07:29.700
The way you can make those match is, you have to remember that F, the joint density function F of Y1 Y2,
00:07:29.700 --> 00:07:36.900
one way to think about that is to sort of first evaluate the marginal density function of Y2.
00:07:36.900 --> 00:07:46.600
And then, once you know what Y2 is, evaluate the conditional density function of Y1 condition on Y2.
00:07:46.600 --> 00:07:52.100
This is that old conditional probability formula.
00:07:52.100 --> 00:08:00.000
If you remember this and then, you kind of plug this into the formula for independence,
00:08:00.000 --> 00:08:13.500
if you plug that in right there, F of Y1 Y2 is equal to F2 of Y2 × F of Y1 condition on Y2.
00:08:13.500 --> 00:08:23.600
What you notice is that from both sides, you can cancel out an F2 of Y2.
00:08:23.600 --> 00:08:38.700
We could cancel and if we canceled F2 of Y2 from both sides, we get on the left F of Y1 condition on Y2.
00:08:38.700 --> 00:08:40.700
We cancel the F2 of Y2.
00:08:40.700 --> 00:08:46.800
On the right, we would get just F1 of Y1.
00:08:46.800 --> 00:08:52.400
That is exactly the intuitive formula that I showed you on the previous side.
00:08:52.400 --> 00:08:58.900
That is how this formula, this definition connects up to the intuitive formula from the previous side.
00:08:58.900 --> 00:09:10.400
This is intuition from the previous slide.
00:09:10.400 --> 00:09:22.100
That is where you can derive the intuitive formula, if you like to have a formal justification
00:09:22.100 --> 00:09:26.900
and how it connects up to the formal definition here.
00:09:26.900 --> 00:09:32.000
There is one last way to think about independence and that comes from a theorem.
00:09:32.000 --> 00:09:34.500
Let me go ahead and show that to you.
00:09:34.500 --> 00:09:42.400
The theorem says that, for continuous random variables Y1 and Y2, they are independent if and only if,
00:09:42.400 --> 00:09:51.800
the domain where the joint density function is defined a non 0, is a rectangle.
00:09:51.800 --> 00:10:01.000
And, the joint density function can be factored into a product of a function of Y1 only and a function of Y2 only.
00:10:01.000 --> 00:10:03.300
Let me expand that a little bit.
00:10:03.300 --> 00:10:14.600
Condition one here means that, when you are graphing the domain, you would have some kind of square or rectangle.
00:10:14.600 --> 00:10:21.500
It could be infinite, you can have something like this, you could have something that goes on forever.
00:10:21.500 --> 00:10:27.400
A rectangle but it goes on forever, maybe something like this where it goes on forever,
00:10:27.400 --> 00:10:38.000
in terms of one variable or it could also go on forever, in terms of the other variable.
00:10:38.000 --> 00:10:43.400
These would all be considered rectangles, even though they extend infinitely far.
00:10:43.400 --> 00:10:49.900
Or even, you can have something that goes on forever in both directions,
00:10:49.900 --> 00:10:54.400
that is still considered to be a rectangle, for the purposes of this theorem.
00:10:54.400 --> 00:11:01.600
What you could not have is some of these triangular domains that we have been looking at, in some of these examples.
00:11:01.600 --> 00:11:06.400
I think we had one example where there was a triangular domain like that.
00:11:06.400 --> 00:11:14.700
That was the triangular domain and that was automatically not independent.
00:11:14.700 --> 00:11:23.200
All these others, at least as far as condition one is concern, would qualify as being independent.
00:11:23.200 --> 00:11:30.900
The second condition means that, you can factor F of the joint density function F of Y1, Y2.
00:11:30.900 --> 00:11:38.800
You can factor that into a function of Y1 × a function of Y2.
00:11:38.800 --> 00:11:44.600
It is okay for either of these functions to be constants meaning, you do not have to see the variables in these functions.
00:11:44.600 --> 00:11:52.500
Sometimes, you might just have a function of Y1 and you would say that other function is just 1,
00:11:52.500 --> 00:12:02.000
and that is still okay to be constant.
00:12:02.000 --> 00:12:05.100
It is okay, if either one of these functions are constants.
00:12:05.100 --> 00:12:14.300
You would have to be able to factor it and separate it into a function of Y1 and a function of Y2 separately.
00:12:14.300 --> 00:12:21.100
Inextricably, next in the density function that you cannot factor it then it is not independent.
00:12:21.100 --> 00:12:22.200
We will work through the examples.
00:12:22.200 --> 00:12:25.600
I’m going to try to solve most of the examples using the definition but then,
00:12:25.600 --> 00:12:30.500
in a lot of them we will come back and applying this theorem.
00:12:30.500 --> 00:12:34.800
We will see that, if we can use this theorem, we could have gotten the answer a lot more quickly,
00:12:34.800 --> 00:12:42.700
just by kind of glancing at the region of definition, or just trying to factor the density function.
00:12:42.700 --> 00:12:47.600
We will try to do the examples both ways, that you can get a feel for both of them.
00:12:47.600 --> 00:12:50.500
Let us jump into those.
00:12:50.500 --> 00:12:55.700
Example 1, this is an example we have seen before in some of the previous videos.
00:12:55.700 --> 00:13:03.800
But, we have not looked at it in quite this light before, in terms of independence.
00:13:03.800 --> 00:13:09.700
We have F of Y1 Y2 is 6 × 1- Y2.
00:13:09.700 --> 00:13:19.300
The region there is a Y1 there, Y2 there, and our region is from 0 to 1 on both variables.
00:13:19.300 --> 00:13:23.600
But, we are only looking at the region where Y2 is bigger than Y1.
00:13:23.600 --> 00:13:31.000
That is this triangular region and we are looking at that color blue region.
00:13:31.000 --> 00:13:34.800
The question is, whether Y1 and Y2 are independent.
00:13:34.800 --> 00:13:40.900
If you are on top of your game right now, you already know the answer because there is a shortcut to the answer.
00:13:40.900 --> 00:13:45.900
I’m going to go ahead and use the definition because that is what the example asks me to do.
00:13:45.900 --> 00:13:49.000
I will use the definition, I will be very honest, I will work it out.
00:13:49.000 --> 00:13:53.900
But kind of secretly that people who already know have already glance of that region and
00:13:53.900 --> 00:13:58.200
there is a shortcut to the answer, that hopefully you already figured it out.
00:13:58.200 --> 00:14:01.200
Let us go ahead and work it out.
00:14:01.200 --> 00:14:07.500
What we are trying to figure out to test our definition of independence is whether F of Y1 Y2,
00:14:07.500 --> 00:14:18.300
the joint density function separates into the two marginal density functions F1 of Y1, F2 of Y2.
00:14:18.300 --> 00:14:22.100
It is a question there, whether those two were equal.
00:14:22.100 --> 00:14:26.300
We will work it out, we will see if they are equal, and then we use that to determine if they are independent.
00:14:26.300 --> 00:14:28.800
That is the definition of independence.
00:14:28.800 --> 00:14:32.000
You got to remember the marginal density function F1 of Y1.
00:14:32.000 --> 00:14:38.100
We did calculate this, this is one of the examples in the previous lecture.
00:14:38.100 --> 00:14:39.600
I will go ahead and calculate it again.
00:14:39.600 --> 00:14:43.900
We always have this variable switch, you always integrate over the other variables.
00:14:43.900 --> 00:14:58.700
This is Y2, and Y2 in this case, goes from the line Y1 = Y2 or Y2 = Y1 to Y2 = 1.
00:14:58.700 --> 00:15:05.500
In this case, we are integrating from Y2 = Y1 to Y2 = 1.
00:15:05.500 --> 00:15:14.000
My joint density function is given in the problem, 6 × 1- Y2.
00:15:14.000 --> 00:15:18.800
We are integrating with respect to Y2.
00:15:18.800 --> 00:15:24.400
I forgot my D in there, that is very important, DY2.
00:15:24.400 --> 00:15:27.300
I will just go ahead and integrate that.
00:15:27.300 --> 00:15:36.200
The integral of 6 is 6Y2 -, the integral of 6Y2 is 3Y2²,
00:15:36.200 --> 00:15:50.300
Y2² integrate that from Y2 = Y1 to Y2 = 1, which is 6 -3, -6Y1.
00:15:50.300 --> 00:15:54.500
That is a Y2, it looks like it did not show up there.
00:15:54.500 --> 00:15:59.000
I forgot my 3Y2², that is very important there.
00:15:59.000 --> 00:16:14.100
-6Y1, - -, + 3Y1² and that simplifies a bit to 3Y1² - 6Y1 + 3.
00:16:14.100 --> 00:16:17.900
Fair enough, that is my marginal density function for Y1.
00:16:17.900 --> 00:16:25.300
F2 of Y2 is, we will switch the roles of the variables there.
00:16:25.300 --> 00:16:33.900
We are integrating over Y1, Y1 goes from 0 up to Y1 = Y2.
00:16:33.900 --> 00:16:38.400
Y1 = 0 to Y1 = Y2.
00:16:38.400 --> 00:16:42.000
I’m doing this a little bit faster than I did in the previous videos.
00:16:42.000 --> 00:16:47.400
We did figure out both of these marginal density functions, as examples in the previous video.
00:16:47.400 --> 00:16:51.200
You can go back and check them out, if you want to see this work out a little more slowly.
00:16:51.200 --> 00:17:00.000
6 × 1- Y2 DY1, I’m integrating with respect to Y1.
00:17:00.000 --> 00:17:03.600
Be very careful here, not with respect to Y2.
00:17:03.600 --> 00:17:10.600
6 × 1- Y2 × Y1, that is because Y2 is just a constant.
00:17:10.600 --> 00:17:18.500
When we integrate with respect to Y1, evaluate that from Y1 = 0 to Y1 = Y2.
00:17:18.500 --> 00:17:24.500
I get 6 × 1- Y2 × Y2.
00:17:24.500 --> 00:17:28.400
Let me look at my condition that I'm trying to check here.
00:17:28.400 --> 00:17:35.600
That is whether the joint density function splits apart into the two marginal density functions.
00:17:35.600 --> 00:17:47.900
That is 6 × 1- Y2 is that equal to F1 of Y1 was this, 3Y1² - 6Y1.
00:17:47.900 --> 00:17:53.400
It looks like I forgot a Y1 up there, + 3.
00:17:53.400 --> 00:18:00.100
That is multiplied by 6 × 1- Y2 × Y2 F2 of Y2.
00:18:00.100 --> 00:18:07.500
Clearly, if we expand out all that mess on the right, we are not going to get the equivalent expression on the left.
00:18:07.500 --> 00:18:10.700
This does not work.
00:18:10.700 --> 00:18:28.200
We can say that the Y1 and Y2 here, these variables Y1 and Y2 are not independent.
00:18:28.200 --> 00:18:36.300
That is a formal check of how to determine whether or not these variables are independent.
00:18:36.300 --> 00:18:42.900
Let me show you the secret shortcut that hopefully you had in mind, even before we started.
00:18:42.900 --> 00:18:53.700
Without doing any calculus at all, I knew that as soon as I graphed this region that this was not independent.
00:18:53.700 --> 00:19:12.100
That is by the theorem, Y1 and Y2 are not independent.
00:19:12.100 --> 00:19:23.200
That is because the theorem said that the variables are independent if and only if, the region is a rectangle.
00:19:23.200 --> 00:19:28.600
Another condition which I do not even have to check because I already know the region is not a rectangle.
00:19:28.600 --> 00:19:41.600
Because the region is not a rectangle, it is a triangle.
00:19:41.600 --> 00:19:49.500
That is another and much quicker way of solving this problem, is to invoke that theorem there.
00:19:49.500 --> 00:19:53.500
That is the two different ways you could solve this problem.
00:19:53.500 --> 00:19:58.000
The problem did ask you to use the definitions, that is why I worked it out from scratch.
00:19:58.000 --> 00:20:02.500
I started with the joint density function and I want to see if it could be,
00:20:02.500 --> 00:20:06.600
if it was really the product of the two marginal density functions.
00:20:06.600 --> 00:20:11.300
I calculated the marginal density function F1 of Y1 and F2 of Y2.
00:20:11.300 --> 00:20:14.900
Each one, you have to switch the variable that you are integrating with respect to.
00:20:14.900 --> 00:20:18.500
F1 of Y1, we integrate with respect to Y2.
00:20:18.500 --> 00:20:22.400
F2 of Y2, we integrate with respect to Y1.
00:20:22.400 --> 00:20:33.200
And then, I describe this region separately, in terms of Y2 or in terms of Y1.
00:20:33.200 --> 00:20:37.300
I ran it through these integral, did a little multivariable calculus.
00:20:37.300 --> 00:20:42.000
And then, I multiply those two marginal density functions together to see whether
00:20:42.000 --> 00:20:45.200
I would get back the joint density function that I started with.
00:20:45.200 --> 00:20:49.200
Actually, I did not even bother to work out the multiplication because I could see that,
00:20:49.200 --> 00:20:53.700
there is no way this is going to come out to be 6 × 1- Y2.
00:20:53.700 --> 00:20:57.900
It is definitely, if you multiply out all this mess on the right, it is not going to work.
00:20:57.900 --> 00:21:06.300
Therefore, by the definition of independence, Y1 and Y2 are not independent.
00:21:06.300 --> 00:21:12.100
A quicker way that we could figure that out is, to use the theorem that I gave you on the third slide of this lecture.
00:21:12.100 --> 00:21:16.400
It just says, first of all, look at the region and see if you got a rectangle.
00:21:16.400 --> 00:21:21.100
If you have not got a rectangle then immediately, you know they are not independent.
00:21:21.100 --> 00:21:23.800
If you have got a rectangle, there is another condition you need to check.
00:21:23.800 --> 00:21:32.600
But, we could have stop as soon as we saw that region was a triangle, we know that they are not independent.
00:21:32.600 --> 00:21:37.300
Let us move on and we are going to look at example 2 now.
00:21:37.300 --> 00:21:45.400
F of Y1 Y2 is defined to be Y1 + Y2 and our region is a square.
00:21:45.400 --> 00:21:47.600
Let me go ahead and graph that out.
00:21:47.600 --> 00:21:53.000
We do not have the easy shortcut that we had on the previous example,
00:21:53.000 --> 00:21:58.200
where we knew that they were not independent because the region was not a rectangle.
00:21:58.200 --> 00:22:00.900
Here, a square counts as being a rectangle.
00:22:00.900 --> 00:22:05.200
Here is Y2, here is Y1, there is 0 for both of them.
00:22:05.200 --> 00:22:10.600
There is 1, there is 1, and so our region is just this very nice square
00:22:10.600 --> 00:22:14.700
which means maybe they are independent because, at least the region is a square.
00:22:14.700 --> 00:22:18.300
But, we are going to use the definition to calculate it out.
00:22:18.300 --> 00:22:22.500
That means, we are going to need to find the two marginal density functions.
00:22:22.500 --> 00:22:27.700
F1 of Y1 means you integrate over Y2.
00:22:27.700 --> 00:22:39.400
It looks like I just integrate for Y2 = 0 to 1, Y2 = 1 of Y1 + Y2 DY2.
00:22:39.400 --> 00:22:46.400
Be careful when you integrate, because you have to integrate keeping in mind that the variable is Y2.
00:22:46.400 --> 00:22:53.200
Y1 is a constant, very common mistake that my students make when they are doing their probability homework,
00:22:53.200 --> 00:22:57.200
is they cannot keep the variable straight, which one you are integrating.
00:22:57.200 --> 00:23:04.000
We integrate Y1, that is a constant, the integral is just Y1 Y2.
00:23:04.000 --> 00:23:08.900
Y2 is the variable, the integral is Y2²/2.
00:23:08.900 --> 00:23:17.500
We want to evaluate all that from Y2 = 0 to Y2 = 1.
00:23:17.500 --> 00:23:27.600
Let us see, when Y2 is 1, we will get Y1 + ½.
00:23:27.600 --> 00:23:30.500
When Y2 is 0, it looks like both the terms dropout.
00:23:30.500 --> 00:23:33.900
I found the marginal density function F1 of Y1.
00:23:33.900 --> 00:23:43.500
F2 of Y2, if you look at the function that we started with, Y1 + Y2 is totally symmetric between Y1 and Y2.
00:23:43.500 --> 00:23:48.300
The region is symmetric too, it is going to be the exact same calculation.
00:23:48.300 --> 00:23:58.700
Just switch the roles of Y1 and Y2, you will end up with Y2 + ½ that is because everything is symmetric in this problem.
00:23:58.700 --> 00:24:01.800
Let me make that a little more clear here, clear that I'm skipping a few steps because
00:24:01.800 --> 00:24:05.900
I can tell that they are going to be the same, as the previous one.
00:24:05.900 --> 00:24:09.700
You are just switching the roles of Y1 and Y2.
00:24:09.700 --> 00:24:13.400
I want to check if Y1 and Y2 are independent.
00:24:13.400 --> 00:24:22.000
I want to check if F of Y1 Y2, the joint density function, is equal to the product
00:24:22.000 --> 00:24:30.300
of the two marginal density functions F1 of Y1 × F2 of Y2.
00:24:30.300 --> 00:24:40.600
In this case, Y1 + Y2 is that equal to Y1 + ½ × Y2 + ½.
00:24:40.600 --> 00:24:44.400
Now, if you multiply those out, no way that is going to be equal.
00:24:44.400 --> 00:24:50.700
It is definitely not going to be equal.
00:24:50.700 --> 00:24:59.700
Y1 and Y2 are not independent, that is the conclusion we have to draw from this.
00:24:59.700 --> 00:25:22.700
Y1 and Y2 are not independent, by the original definition of independence.
00:25:22.700 --> 00:25:28.100
The way I calculated that was, I really wanted to check, here is the definition of independence right here.
00:25:28.100 --> 00:25:33.500
It says that the joint density function is equal to the product of the marginal density functions.
00:25:33.500 --> 00:25:40.500
But I worked out the marginal density functions, F1 of Y1 means you integrate with respect to Y2.
00:25:40.500 --> 00:25:47.900
I integrated the joint density function, I had to be careful there that Y2 was the variable and Y1 was just a constant.
00:25:47.900 --> 00:25:52.400
That is why I got Y1 × Y2 here and Y2².
00:25:52.400 --> 00:25:56.700
Worked out to Y1 + ½, it is a function of Y1.
00:25:56.700 --> 00:26:02.100
Y2 works the exact same way, it gives you Y2 + ½.
00:26:02.100 --> 00:26:07.500
When I plug those in, Y1 + ½ × Y2 + ½, if you multiply those together,
00:26:07.500 --> 00:26:13.600
will you get the original joint density function that we start out with, no you do not get that.
00:26:13.600 --> 00:26:15.400
They are not independent.
00:26:15.400 --> 00:26:19.200
By the way, it is less obvious than it was in example 1.
00:26:19.200 --> 00:26:22.000
In example 1, we had our region that was a triangle.
00:26:22.000 --> 00:26:26.700
Immediately, the theorem told you that it was not independent.
00:26:26.700 --> 00:26:32.300
In this case, our region was a square and that condition did not make it obvious anymore.
00:26:32.300 --> 00:26:39.700
Maybe, you could have looked at Y1 + Y2 and said, can I factor that into a function of Y1 × a function of Y2.
00:26:39.700 --> 00:26:46.000
And said, you cannot factor that, then you would have known that they are not independent.
00:26:46.000 --> 00:26:51.700
The safest way is actually to check this definition and to calculate the marginal density functions,
00:26:51.700 --> 00:26:55.900
and see if they multiply to the joint density function.
00:26:55.900 --> 00:27:02.900
They do not, in this case, the variables are not independent.
00:27:02.900 --> 00:27:07.400
In example 3, we have a discreet situation.
00:27:07.400 --> 00:27:11.700
We are going to roll two dice, a red dice and a blue dice.
00:27:11.700 --> 00:27:19.400
We are going to define the variables Y1 is what shows on the red dice and Y2 is the total.
00:27:19.400 --> 00:27:23.400
You might think, Y1 be what shows on the red dice and Y2 is the blue dice.
00:27:23.400 --> 00:27:28.200
We mixed them up a little bit to make it a little more interesting.
00:27:28.200 --> 00:27:32.000
The question is, whether Y1 and Y2 are independent.
00:27:32.000 --> 00:27:39.400
Let me just mention that there is an intuitive answer to this, which should make sense to you.
00:27:39.400 --> 00:27:47.500
Intuition here is that, remember the intuition of independence is that if I tell you the value of one of the variables,
00:27:47.500 --> 00:27:52.300
you will have some new information about the other.
00:27:52.300 --> 00:27:57.200
In particular, this Y1, we know it is going to be somewhere between 1 and 6.
00:27:57.200 --> 00:28:04.600
Y2 is going to be somewhere between 2 and 12 because it is the total showing on both dice.
00:28:04.600 --> 00:28:17.400
The intuition is, if I tell you what is showing on one of the dice, or if I tell you what one of the variables is,
00:28:17.400 --> 00:28:21.000
does it change what you might expect about the others, about the other one?
00:28:21.000 --> 00:28:23.300
In this case, yes, it does.
00:28:23.300 --> 00:28:33.200
Intuition is, it does change your prediction, that means these variables are dependent on each other,
00:28:33.200 --> 00:28:35.700
they are not independent.
00:28:35.700 --> 00:28:38.100
Let me write that down to make it clear.
00:28:38.100 --> 00:28:49.200
No, they are not independent because, let me just give an example value here.
00:28:49.200 --> 00:28:55.500
Let me say, suppose you roll these two dice and you are wondering what kind of roll you are going to get.
00:28:55.500 --> 00:29:05.700
Suppose, you peeked at the red dice, if you get a 6, if Y1 = 6 that means you peek at the red dice and
00:29:05.700 --> 00:29:10.900
say what dice came out to be a 6?
00:29:10.900 --> 00:29:15.200
My prediction is, I’m more likely to have a high total.
00:29:15.200 --> 00:29:18.800
That is going to change what I expect about Y2.
00:29:18.800 --> 00:29:35.000
Then, Y2 is more likely to be large.
00:29:35.000 --> 00:29:40.400
If I know that one die rolled very high, then, it is more likely that I got a large total.
00:29:40.400 --> 00:29:46.300
In particular, if I just say I'm rolling two dice, I could get anything from 2 to 12.
00:29:46.300 --> 00:29:51.800
But if you tell me that Y1 is 6, I know I'm not going to get 2 as a total, not anymore.
00:29:51.800 --> 00:29:54.300
I know that I’m going to get at least 6 as the total.
00:29:54.300 --> 00:30:04.700
That is a very strong intuitive hint that these variables do depend on each other, they are not independent.
00:30:04.700 --> 00:30:08.500
Let me check it using the formulas as well.
00:30:08.500 --> 00:30:14.600
I'm going to check P of Y1 Y2, this is the definition of independent.
00:30:14.600 --> 00:30:26.300
It should be equal to P1 of Y1 × P2 of Y2.
00:30:26.300 --> 00:30:31.000
I do not know whether that is true, if they are independent then they should be true.
00:30:31.000 --> 00:30:36.900
I'm going to take some values of Y1 and Y2, I will go ahead and take those values that I mentioned.
00:30:36.900 --> 00:30:41.400
Y1 is equal to 6, I’m going to pick those.
00:30:41.400 --> 00:30:44.100
This formula should be true for all values.
00:30:44.100 --> 00:30:50.400
If it is not going to be true, I can pick whatever values I want to illustrate that it is not true.
00:30:50.400 --> 00:30:57.100
Y2, I will pick 12 just because I think that, if I know the red dice is 6,
00:30:57.100 --> 00:30:59.800
I think that is going to change my probability of getting a 12.
00:30:59.800 --> 00:31:13.300
Just think about whether the probability of 6/12 is equal to P1 of 6 × P2 of 12.
00:31:13.300 --> 00:31:20.300
The probability of 6/12 means that I got a 6 on the red dice and a 12 total.
00:31:20.300 --> 00:31:25.300
In order to get that, I have to get a 6 on the red and a 6 on the blue.
00:31:25.300 --> 00:31:30.500
This is really the probability of 6-6.
00:31:30.500 --> 00:31:37.900
The probability of rolling double 6 is 1/36, 1/6 × 1/6.
00:31:37.900 --> 00:31:46.900
P1 of 6, what is my probability that the red dice is equal to 6, that is 1/6.
00:31:46.900 --> 00:31:52.100
P2 of 12, what is my probability that my total is 12?
00:31:52.100 --> 00:32:00.400
Again, to get 12, I have to get 6 on both dice, that is 1/36.
00:32:00.400 --> 00:32:08.300
Now, is 1/36 equal to 1/6 × 1/36, sure is not.
00:32:08.300 --> 00:32:11.200
That does not work out.
00:32:11.200 --> 00:32:18.200
Since, we found some values for which that equality did not hold, we can say for sure,
00:32:18.200 --> 00:32:23.800
that Y1 and Y2 are not independent.
00:32:23.800 --> 00:32:31.200
That agrees with the intuitive answer that we already gave.
00:32:31.200 --> 00:32:37.000
That does agree with the intuition, that is quite reassuring that our intuition is not completely off base
00:32:37.000 --> 00:32:39.700
and the formulas do back it up.
00:32:39.700 --> 00:32:42.000
Let me recap that.
00:32:42.000 --> 00:32:47.200
We are rolling two dice, we have, what shows on the red dice and the total.
00:32:47.200 --> 00:32:49.400
The question is, whether those are independent.
00:32:49.400 --> 00:32:56.400
I do not think they are going to be independent because I think, if you to tell me what is going to come up on the red dice,
00:32:56.400 --> 00:32:59.800
then I can probably say a little more about what the total is likely to be.
00:32:59.800 --> 00:33:03.800
I will not be able to say exactly, but if you tell me that I get a 6 on the red dice,
00:33:03.800 --> 00:33:08.300
then I know the total is somewhere between 7 and 12.
00:33:08.300 --> 00:33:13.000
If you tell me that I get a 1 on the red dice, then I know the total is somewhere between 2 and 7.
00:33:13.000 --> 00:33:16.900
It is really going to change, what I expect the total to be.
00:33:16.900 --> 00:33:22.700
Similarly, if you tell me what the total is, maybe, I know a little more about what the red dice might be showing.
00:33:22.700 --> 00:33:27.500
Like, if you tell me that the total is 12, I know the red dice is a 6 now.
00:33:27.500 --> 00:33:36.500
That is the intuition there, which is that knowing one variable will influence what you predict for the other variable.
00:33:36.500 --> 00:33:41.900
That means they are dependent on each other, which means they are not independent.
00:33:41.900 --> 00:33:44.800
That is why I made that intuitive prediction.
00:33:44.800 --> 00:33:47.300
In order to back it up, I checked it with the formula.
00:33:47.300 --> 00:33:58.100
I just grabbed two values of Y1 and Y2, if they are independent then this formula should hold for all values of Y1 and Y2.
00:33:58.100 --> 00:34:01.900
That is my definition of independence.
00:34:01.900 --> 00:34:07.800
I'm going to check it out just with these two values, the Y1 and Y2, 6 and 12.
00:34:07.800 --> 00:34:14.800
On the left, I’m finding the probability that the red dice is 6 and the total is 12 which means,
00:34:14.800 --> 00:34:17.700
we must have rolled double 6.
00:34:17.700 --> 00:34:21.300
The chance of getting a double 6 is 1/36.
00:34:21.300 --> 00:34:28.200
On the right, P1 of 6 means what is the probability that the red dice is a 6, it is 1/6.
00:34:28.200 --> 00:34:30.500
What is the probability that the total is 12?
00:34:30.500 --> 00:34:34.800
Again, you would take double 6, that is 1/36.
00:34:34.800 --> 00:34:37.900
I just check if the arithmetic works out and it does not.
00:34:37.900 --> 00:34:42.000
1/36 is not equal to 1/6 × 1/36.
00:34:42.000 --> 00:34:53.500
Y1 and Y2 are not independent and that confirms the intuitively that I made, at the beginning of the example.
00:34:53.500 --> 00:35:02.900
In example 4, we have the joint density function F of Y1 Y2 is E ⁻Y1 + Y2.
00:35:02.900 --> 00:35:16.400
My region here is the region on Y1 and Y2 both going from 0 to infinity.
00:35:16.400 --> 00:35:22.900
There is Y1 and there is Y2, both regions go from 0 to infinity.
00:35:22.900 --> 00:35:26.700
The question is, are Y1 and Y2 independent?
00:35:26.700 --> 00:35:29.900
Again, there is sort of two ways I can check this.
00:35:29.900 --> 00:35:36.000
One, is by using the original definition of independence.
00:35:36.000 --> 00:35:39.600
And one, is by using the theorem that we got.
00:35:39.600 --> 00:35:41.800
The faster way will actually be the theorem.
00:35:41.800 --> 00:35:46.800
I'm going to check it from the definition first, just so that you understand that method.
00:35:46.800 --> 00:35:51.100
And then, we will see how the theorem would actually be much faster.
00:35:51.100 --> 00:35:53.300
We will check it out using the definition.
00:35:53.300 --> 00:36:01.600
Remember, the definition of independence was that F of Y1 Y2 should be equal to,
00:36:01.600 --> 00:36:10.400
should factor into the marginal density functions F1 of Y1 and F2 of Y2.
00:36:10.400 --> 00:36:14.200
Let me work out what those are, those marginal density functions.
00:36:14.200 --> 00:36:19.200
F1 of Y1, by definition, that means you switch the variable.
00:36:19.200 --> 00:36:23.900
F1 of Y1, I was already thinking ahead to the integral that I’m about to solve.
00:36:23.900 --> 00:36:32.200
I have to integrate over Y2, Y2= 0 to infinity, Y2 goes to infinity.
00:36:32.200 --> 00:36:44.600
I will take a limit for that, of the joint density function E ^ (-Y1 + Y2) DY2.
00:36:44.600 --> 00:36:50.500
When I solve that out, I’m integrating with respect to Y2.
00:36:50.500 --> 00:36:59.200
If I think about that, I can factor out an E ^- Y2 and an E ^- Y1.
00:36:59.200 --> 00:37:05.500
E ⁻Y1 will just be a constant, factor that right on out.
00:37:05.500 --> 00:37:13.200
That is E ⁻Y1, I got the integral of E ⁻Y2 DY2.
00:37:13.200 --> 00:37:21.500
And that is not such a bad integral, it is E ⁻Y1 × –E ⁻Y2.
00:37:21.500 --> 00:37:32.300
I want to evaluate that from Y to = 0, and then take the limit as Y2 goes to infinity, that is E ⁻Y1.
00:37:32.300 --> 00:37:45.100
Y2 going to infinity means, we are talking about E ⁻infinity, that is 1/E ⁺infinity or 1/infinity which is just 0.
00:37:45.100 --> 00:37:57.000
-E⁻⁰, -E⁰ which is 1, those two negatives cancel and I just get a +1.
00:37:57.000 --> 00:38:03.300
I get E ⁻Y1, notice that I get a function of Y1 which is what I'm supposed to get,
00:38:03.300 --> 00:38:08.700
when I take the marginal density function.
00:38:08.700 --> 00:38:14.500
This is actually symmetric, F2 of Y2 was going to behave the exact same way.
00:38:14.500 --> 00:38:20.400
I’m not going to belabor the details there, it is going to be E ⁻Y2.
00:38:20.400 --> 00:38:26.000
I will check out my definition of independence, F of Y1 Y2.
00:38:26.000 --> 00:38:35.700
Y2 is equal to, or possibly equal to F1 of Y1 × F2 of Y2.
00:38:35.700 --> 00:38:43.100
E ⁻Y1 + Y2, that is my joint density function that I have been given.
00:38:43.100 --> 00:38:47.200
F1 of Y1, I worked out was E ⁻Y1.
00:38:47.200 --> 00:38:53.700
F2 of Y2, I worked out was E ⁻Y2.
00:38:53.700 --> 00:38:57.800
Those could combine, we get E ^- Y1 + Y2.
00:38:57.800 --> 00:39:06.300
In fact, it does work, that equality really holds true.
00:39:06.300 --> 00:39:20.300
By the definition, yes, they are independent.
00:39:20.300 --> 00:39:24.600
Y1 and Y2 are independent, checking from the definitions.
00:39:24.600 --> 00:39:28.400
That is really very reassuring there.
00:39:28.400 --> 00:39:39.200
Let me show you another way you could have done this problem, which is to use the theorem, two parts of the theorem.
00:39:39.200 --> 00:39:43.900
By the way, I gave you this theorem in the third slide of these lectures.
00:39:43.900 --> 00:39:46.100
Just check back and you will see the third slide.
00:39:46.100 --> 00:39:56.000
The domain is a rectangle, it is an infinite rectangle.
00:39:56.000 --> 00:39:59.000
But, that does check out here.
00:39:59.000 --> 00:40:06.200
Remember, it is okay for it to be infinite, according to the theorem, it is okay if the rectangle is infinite.
00:40:06.200 --> 00:40:09.800
I'm looking at this domain here, it is an infinite rectangle.
00:40:09.800 --> 00:40:15.400
That is condition one of the theorem, that is satisfied.
00:40:15.400 --> 00:40:23.200
Condition two of the theorem was that, the joint density function F of Y1 Y2
00:40:23.200 --> 00:40:28.600
had to factor into a function of Y1 only × the function of Y2 only.
00:40:28.600 --> 00:40:31.100
Let us check that out here.
00:40:31.100 --> 00:40:52.700
E ⁻Y1 + Y2, yes, I can factor that into E ⁻Y1 × E ⁻Y2 which is a function of, let me just write that as G of Y1.
00:40:52.700 --> 00:41:03.800
Y1 × H of Y2 because I do have Y1 only in the first function and Y2 only in the second function.
00:41:03.800 --> 00:41:08.500
That second edition of the theorem is satisfied.
00:41:08.500 --> 00:41:15.500
Once, I have checked both of those conditions, I can go to that same conclusion and say yes, they are independent.
00:41:15.500 --> 00:41:20.700
I could have saved myself doing a lot of integration there, if I had used the theorem.
00:41:20.700 --> 00:41:23.800
I want to make sure that you are comfortable using the definition.
00:41:23.800 --> 00:41:29.400
But also, using the theorem which can save you lots of time, if you know how to use it.
00:41:29.400 --> 00:41:30.800
Let me recap the steps here.
00:41:30.800 --> 00:41:35.300
The first way that I want to check this problem was to look at the definition.
00:41:35.300 --> 00:41:41.400
Does the joint density function factor into the two marginal density functions?
00:41:41.400 --> 00:41:48.300
I had to calculate the two marginal density functions F1 of Y1, you switch the variables and
00:41:48.300 --> 00:41:52.200
you integrate over Y2 with respect to Y2.
00:41:52.200 --> 00:41:57.300
I looked at my range on Y2, that goes from 0 to infinity.
00:41:57.300 --> 00:42:00.500
That is where I got these limits right here.
00:42:00.500 --> 00:42:06.400
I integrated the joint density function that I was given, E ⁻Y1 + Y2.
00:42:06.400 --> 00:42:09.500
That is where that came from.
00:42:09.500 --> 00:42:18.400
That factors, it is really nice that if factors because we are integrating with respect to Y2, that means the term with Y1,
00:42:18.400 --> 00:42:24.500
pulls right out of the integral and then I'm just doing an integral on Y2, pretty easy one.
00:42:24.500 --> 00:42:32.800
Plug in my limits, infinity and 0, and I just simplify down to E ⁻Y1.
00:42:32.800 --> 00:42:38.000
That was the marginal density function on Y1.
00:42:38.000 --> 00:42:46.100
The exact same arithmetic occurs with Y2, except you are just switching the two variables.
00:42:46.100 --> 00:42:57.800
I'm going to check that definition, does the joint density function separate into the two marginal density functions.
00:42:57.800 --> 00:43:01.800
When I plugged everything in there, it look like I had a true equation.
00:43:01.800 --> 00:43:07.400
Just by the definition, I get that the two variables are independent.
00:43:07.400 --> 00:43:13.300
A quicker way to do that would have been, both the theorem from the third slide of this lecture.
00:43:13.300 --> 00:43:16.900
But we look at the domain, that is looking at this domain right here.
00:43:16.900 --> 00:43:25.200
It is an infinite rectangle, that does satisfy it is not a triangular region or anything like that.
00:43:25.200 --> 00:43:32.200
If you look at the joint density function, we can factor it into a function of Y1 × a function of Y2.
00:43:32.200 --> 00:43:35.100
They separate the variables there.
00:43:35.100 --> 00:43:45.500
And that, right out there, would have been enough to confirm to me that these variables really are independent.
00:43:45.500 --> 00:43:48.800
In our last example here on independent random variables,
00:43:48.800 --> 00:43:57.200
we are given a joint density function of 4Y1 Y2 and our region, I will go ahead and graph it.
00:43:57.200 --> 00:44:03.900
Y1 and Y2 both going from 0 to 1, here is Y2 and here is Y1.
00:44:03.900 --> 00:44:11.800
We want to figure out, I seem to switch my variables for some reason, I certainly would not want do that.
00:44:11.800 --> 00:44:19.900
There is Y1 and there is Y2, here is 0 and 1 on both axis.
00:44:19.900 --> 00:44:22.800
There is my region, a very nice square.
00:44:22.800 --> 00:44:26.200
By the way, if you are really on your game right now,
00:44:26.200 --> 00:44:32.000
if you have been paying close attention to everything in this lecture, you already know the answer to this problem.
00:44:32.000 --> 00:44:34.000
If you really know what is going on.
00:44:34.000 --> 00:44:39.700
I'm going to take it a little slowly and we will work it through.
00:44:39.700 --> 00:44:45.300
We will find an answer and then I will kind of comeback at the end, and show you how you could have done it very quickly.
00:44:45.300 --> 00:44:49.800
If you knew quickly, what you are doing.
00:44:49.800 --> 00:44:54.800
We are going to check the definition of independence which it asks us
00:44:54.800 --> 00:45:00.400
whether the joint density function factors into the product of the marginal densities.
00:45:00.400 --> 00:45:08.000
Let me find the marginal densities, F1 of Y1 is equal to, we integrate on Y2 here.
00:45:08.000 --> 00:45:18.400
Y2 goes from 0 to 1, my joint density function is 4Y1 Y2, and we are integrating DY2.
00:45:18.400 --> 00:45:27.200
If I integrate that, the Y1 is a constant and the integral of 4Y2 is 2Y2².
00:45:27.200 --> 00:45:31.700
2Y2², we still have the constant Y1.
00:45:31.700 --> 00:45:39.100
We integrate that or we evaluate that from Y2 = 0 to Y2 = 1.
00:45:39.100 --> 00:45:44.700
If I plug in Y2 = 1, I just get 2Y1.
00:45:44.700 --> 00:45:46.800
Y2 = 0 does mean nothing.
00:45:46.800 --> 00:45:50.300
I have figured out my marginal density function F1 of Y1.
00:45:50.300 --> 00:45:59.000
F2 of Y2, if I do this exact same arithmetic, everything is symmetric here.
00:45:59.000 --> 00:46:01.000
We will just swap the value of the variables.
00:46:01.000 --> 00:46:08.400
It is going to work out to be 2Y2, that would be the marginal density function for Y2.
00:46:08.400 --> 00:46:18.600
I want to check using the definition of independence, is F of Y1 Y2 is it equal to F1 of Y1,
00:46:18.600 --> 00:46:22.800
the thing that I just calculated × F2 of Y2.
00:46:22.800 --> 00:46:24.000
Is that going to work out?
00:46:24.000 --> 00:46:26.900
I will plug in everything and I will see if it works out.
00:46:26.900 --> 00:46:37.700
F4 × Y1 Y2 is that equal to 2Y1, that is what I just calculated, × 2Y2.
00:46:37.700 --> 00:46:48.500
That was still a question, but when I look at them, it is really is, 4Y1 Y2 is 2 × Y1 × 2 × Y2.
00:46:48.500 --> 00:46:50.800
Low and behold, it does check.
00:46:50.800 --> 00:47:06.100
I really confirmed by the definition that, yes, Y1 and Y2 are independent.
00:47:06.100 --> 00:47:10.100
That is very reassuring, that they do come out to be independent.
00:47:10.100 --> 00:47:16.400
However, I hope that some of you watching the video were kind of chuckling to yourselves all along,
00:47:16.400 --> 00:47:18.900
because you knew this answer in advance.
00:47:18.900 --> 00:47:21.600
Here is how you knew the answer in advance.
00:47:21.600 --> 00:47:27.700
You remembered that theorem that I gave you on the third slide of the lecture.
00:47:27.700 --> 00:47:29.500
You can go back and check that out.
00:47:29.500 --> 00:47:32.000
There are two conditions you had to check.
00:47:32.000 --> 00:47:41.000
The domain is a rectangle and a square, in this case, does qualify as a rectangle.
00:47:41.000 --> 00:47:46.800
You checked right away that the domain is a rectangle, that is confirmed.
00:47:46.800 --> 00:47:51.500
The second condition that you have to check is that the joint density function,
00:47:51.500 --> 00:48:03.000
F of Y1 Y2 factors into a function of Y1 × Y2.
00:48:03.000 --> 00:48:06.400
Let me go ahead and say it is 4Y1 Y2.
00:48:06.400 --> 00:48:12.700
If you wanted, you could write that as 4Y1 × Y2.
00:48:12.700 --> 00:48:15.400
Or you can put a 2 on each part, it really does not matter.
00:48:15.400 --> 00:48:23.500
The important thing here is that it is a function of Y1 only, × a function of Y2 only.
00:48:23.500 --> 00:48:31.100
It does indeed factor as it is supposed to, in order to be independent.
00:48:31.100 --> 00:48:38.100
Both of those are probably things you could have checked in your head, if you really knew what was going on.
00:48:38.100 --> 00:48:45.500
In that case, you knew right away from the beginning of the problem, by the theorem Y1 and Y2 are independent.
00:48:45.500 --> 00:48:53.600
You could save yourself this integration and the tedious checking, and jump to the answer right away.
00:48:53.600 --> 00:48:55.600
Let me recap that problem.
00:48:55.600 --> 00:48:59.000
I wanted to check this definition of independence,
00:48:59.000 --> 00:49:05.100
that the joint density function does occur as the product of the two marginal density functions.
00:49:05.100 --> 00:49:09.400
That meant, I had to calculate the two marginal density functions.
00:49:09.400 --> 00:49:15.900
F1 of Y1, to calculate the marginal density function, you integrate over the other variable.
00:49:15.900 --> 00:49:23.100
I put my Y2 there and my region is Y2 goes from 0 to 1.
00:49:23.100 --> 00:49:27.400
That is how I got those limits right there, Y2 goes from 0 to 1.
00:49:27.400 --> 00:49:35.700
My joint density function is 4Y1 Y2, I got that from the stem of the problem there.
00:49:35.700 --> 00:49:39.300
Integrate that with respect to Y2.
00:49:39.300 --> 00:49:46.500
The Y1 just comes along as a constant, the integral of 4Y2 is 2Y2².
00:49:46.500 --> 00:49:50.600
Plug in the values for Y2, I get 2Y1.
00:49:50.600 --> 00:49:54.400
The F2 of Y2 is completely symmetric.
00:49:54.400 --> 00:49:57.600
The function and the region are both symmetric.
00:49:57.600 --> 00:50:01.200
It is going to work out to be 2Y2.
00:50:01.200 --> 00:50:06.300
And then, if I plug those both in, the 2Y1 and the 2Y2, I multiply them together and look,
00:50:06.300 --> 00:50:11.000
I really do get the original joint density function that we started with.
00:50:11.000 --> 00:50:13.500
It really did work out, in this example.
00:50:13.500 --> 00:50:18.000
This is kind of a special example, if you check back some of the previous examples, example 1 and 2,
00:50:18.000 --> 00:50:20.200
it did not work out when we multiply those.
00:50:20.200 --> 00:50:26.100
In this case, it did work out and we can say that the variables are independent.
00:50:26.100 --> 00:50:31.800
The quicker way to do that is to use the theorem that we had in the third slide of this lecture,
00:50:31.800 --> 00:50:35.400
is to just look at this domain right here, and say that is a square.
00:50:35.400 --> 00:50:38.100
A square counts as being a rectangle.
00:50:38.100 --> 00:50:43.900
And then, you look quickly at the joint density function and say can I factor it somehow,
00:50:43.900 --> 00:50:49.000
with all the Y1 on one side and all the Y2 in the other factor.
00:50:49.000 --> 00:50:55.600
Yes I can, I can factor it just like this, all the Y1 in one part and all the Y2 on the other part.
00:50:55.600 --> 00:51:02.100
I have successfully separated it into a function of Y1 × the function of Y2.
00:51:02.100 --> 00:51:05.000
And that was the second condition that you had to check with the theorem.
00:51:05.000 --> 00:51:11.300
The theorem says that both of those conditions are met, then Y1 and Y2 really are independent.
00:51:11.300 --> 00:51:18.300
That is by far, a faster way to know quickly whether your variables are independent.
00:51:18.300 --> 00:51:23.000
That wraps up this lecture on independent random variables.
00:51:23.000 --> 00:51:30.500
This is part of the larger chapter on Bivariate distributions and Bivariate density functions.
00:51:30.500 --> 00:51:36.100
In turn, that is part of the probability videos here on www.educator.com.
00:51:36.100 --> 00:51:39.000
My name is Will Murray and I thank you very much for joining me today, bye now.