WEBVTT mathematics/probability/murray
00:00:00.000 --> 00:00:04.500
Hello, and welcome to the probability lectures here on www.educator.com.
00:00:04.500 --> 00:00:08.800
We are starting a chapter now on continuous distributions.
00:00:08.800 --> 00:00:12.000
My name is Will Murray and I’m your guide today.
00:00:12.000 --> 00:00:17.600
First section here is on density and cumulative distribution functions.
00:00:17.600 --> 00:00:26.600
Anytime you have a continuous probability distribution, you will have a density function and a cumulative distribution function.
00:00:26.600 --> 00:00:32.000
I’m going to explain what those are and make sure that you do not get those 2 mixed up.
00:00:32.000 --> 00:00:39.100
We use F for both of them, that is always a little confusing but I will try to highlight the differences there,
00:00:39.100 --> 00:00:43.400
and then we will do some example problems to practice them.
00:00:43.400 --> 00:00:45.900
Let us jump in with density functions.
00:00:45.900 --> 00:00:49.000
Y is a continuous random variable.
00:00:49.000 --> 00:00:56.100
What that means is it contained values over a whole continuum of possibilities.
00:00:56.100 --> 00:01:05.700
Instead of having discrete probability were Y would be a whole number or a certain number of possible values,
00:01:05.700 --> 00:01:10.500
the values now for Y can be a whole range of things.
00:01:10.500 --> 00:01:17.500
Things like the normal distribution would be a typical example of a continuous random variable.
00:01:17.500 --> 00:01:20.800
The values of Y can be anything.
00:01:20.800 --> 00:01:24.800
It can be any real number at this point.
00:01:24.800 --> 00:01:28.900
It has a density function that, we are going to use f for density functions.
00:01:28.900 --> 00:01:34.700
Be very careful here to distinguish between the density function and what we are going to learn next,
00:01:34.700 --> 00:01:37.400
which is the cumulative distribution function.
00:01:37.400 --> 00:01:45.800
We will use f for the density function and we will use F for the cumulative distribution function.
00:01:45.800 --> 00:01:53.400
If you have sloppy handwriting, now is the time to be very careful to be clear about the difference between
00:01:53.400 --> 00:02:01.500
f of Y which is the density function, and F of Y which is the cumulative distribution function.
00:02:01.500 --> 00:02:03.100
We will learn about that in the next slide.
00:02:03.100 --> 00:02:06.900
Be very careful about the difference between those two.
00:02:06.900 --> 00:02:12.100
The density function, the properties it has to satisfy, it is always positive.
00:02:12.100 --> 00:02:20.000
Essentially that means you cannot have negative probabilities, the probability is always positive.
00:02:20.000 --> 00:02:31.200
The total amount of area under this graph from -infinity to infinity is always exactly equal to 1,
00:02:31.200 --> 00:02:34.900
that is because it is a probability function.
00:02:34.900 --> 00:02:38.700
The total probability of something happening has to be equal to 1.
00:02:38.700 --> 00:02:43.300
Something has to happen and it happens with probability 1.
00:02:43.300 --> 00:02:49.900
The way you use the density function to calculate probabilities is, you always talk in terms of ranges.
00:02:49.900 --> 00:02:56.200
You never talk about the probability of Y being equal to a specific value.
00:02:56.200 --> 00:02:59.700
Back when we are talking about discreet distributions, like the Poisson distribution,
00:02:59.700 --> 00:03:06.800
the binomial distribution, geometric distribution, we would say what is the probability that Y is equal to 3?
00:03:06.800 --> 00:03:11.600
What is the probability that there will be exactly 3 forest fires next year in California?
00:03:11.600 --> 00:03:16.300
What is the probability that the coin will come up heads up 3 ×?
00:03:16.300 --> 00:03:21.300
We would ask what is the probability that Y is equal to a particular value?
00:03:21.300 --> 00:03:23.600
Now, we ask about ranges.
00:03:23.600 --> 00:03:32.500
We will never ask the probability that Y is equal to 3, that would just be 0 because we have so many possible values.
00:03:32.500 --> 00:03:40.400
Instead, we will ask about what is the probability that Y is between one number and another?
00:03:40.400 --> 00:03:45.600
For example, we will find a probability that Y is between A and B.
00:03:45.600 --> 00:03:52.300
The way you find that probability is, you calculate the area under the density function.
00:03:52.300 --> 00:04:00.100
In order to calculate that area, what you do is you take an integral.
00:04:00.100 --> 00:04:14.700
The probability that Y is between constant values A and B is the integral of the density function F of Y DY from Y = a to Y = b.
00:04:14.700 --> 00:04:18.300
That is how you calculate probabilities from now on.
00:04:18.300 --> 00:04:21.800
Remember, that is the density function, that is not to be confused with
00:04:21.800 --> 00:04:29.700
the cumulative distribution function which is the next thing we are going to learn about.
00:04:29.700 --> 00:04:37.000
The cumulative distribution function that is closely related to the density function but it is not the same thing.
00:04:37.000 --> 00:04:40.900
The cumulative distribution function we use F of Y.
00:04:40.900 --> 00:04:43.300
Remember, the density function was f.
00:04:43.300 --> 00:04:54.700
The cumulative distribution function is the probability that Y is less than or equal to a particular cutoff value of Y.
00:04:54.700 --> 00:04:56.800
Let me show you how you find that.
00:04:56.800 --> 00:05:02.800
If this is the density function that I'm graphing right now, this is f of Y.
00:05:02.800 --> 00:05:10.600
You have a particular cutoff value of Y and you want to find the probability of being less than that value.
00:05:10.600 --> 00:05:17.000
The way you do it is you calculate the area to the left of that cutoff.
00:05:17.000 --> 00:05:19.200
You can do that as an integral.
00:05:19.200 --> 00:05:25.600
We already used y as the cutoff, I cannot use my Y as my variable of integration.
00:05:25.600 --> 00:05:30.300
I cannot use of Y, I’m using T here, F of T DT.
00:05:30.300 --> 00:05:36.100
By the way, that is a very common mistake that I see students make in doing probability problems is,
00:05:36.100 --> 00:05:38.300
they will mix up their variables.
00:05:38.300 --> 00:05:42.400
They will have a Y in here and then they will also have that Y over there in the limit.
00:05:42.400 --> 00:05:46.500
That is very bad practice, you can get yourself in lots of problems that way.
00:05:46.500 --> 00:05:52.500
Do not do that, use T when you are calculating the cumulative distribution function.
00:05:52.500 --> 00:05:59.400
Use T as your variable of integration, and then use Y as your limit.
00:05:59.400 --> 00:06:05.200
We can also use F, the cumulative distribution function to calculate probabilities.
00:06:05.200 --> 00:06:12.700
The probability that you are within one range between A and B.
00:06:12.700 --> 00:06:18.300
Let me graphically illustrate this.
00:06:18.300 --> 00:06:26.200
The probability that you are between A and B.
00:06:26.200 --> 00:06:28.900
That is the area in between A and B.
00:06:28.900 --> 00:06:42.500
One way of calculating that is to calculate all the area less than B, and then to subtract off all the area that is less than A.
00:06:42.500 --> 00:06:46.100
To subtract off all this area less than A.
00:06:46.100 --> 00:06:53.700
What you are left with is exactly the area that you want, which is the area between A and B.
00:06:53.700 --> 00:07:00.800
The way you calculate the area that is less than B is to use the cumulative distribution function F of B.
00:07:00.800 --> 00:07:04.200
The area less than A is F of A.
00:07:04.200 --> 00:07:09.500
Once you worked out what F is, you do not have to do any more integration.
00:07:09.500 --> 00:07:12.300
You just plug in your limits B and A.
00:07:12.300 --> 00:07:17.500
Essentially, this is the fundamental theorem of calculus coming through in a probability setting.
00:07:17.500 --> 00:07:26.400
It means once you have done this integral, you just plug in the 2 limits F of B and F of A.
00:07:26.400 --> 00:07:30.700
We are going to study some properties of the cumulative distribution function,
00:07:30.700 --> 00:07:35.200
that is what CDF stand for cumulative distribution function.
00:07:35.200 --> 00:07:45.500
What today's functions look like, remember that if the density function often looks like something like this.
00:07:45.500 --> 00:07:48.500
This is the density function, f of Y.
00:07:48.500 --> 00:07:58.400
The cumulative distribution function represents the area underneath or to the left of any given cutoff.
00:07:58.400 --> 00:08:07.900
The cumulative distribution function therefore, as you start in the left hand side of the universe, at –infinity,
00:08:07.900 --> 00:08:09.800
there is no area to the left.
00:08:09.800 --> 00:08:14.700
It always has to start at 0, let me draw that in black here.
00:08:14.700 --> 00:08:24.600
It always has to start at 0, that is why F of -infinity is always 0.
00:08:24.600 --> 00:08:31.100
What I’m doing here is I’m going to graph F of Y.
00:08:31.100 --> 00:08:43.100
As you start to increase Y here, you get more and more area until you get to the right hand edge of the universe at infinity.
00:08:43.100 --> 00:08:46.600
You got all the area under the density function.
00:08:46.600 --> 00:08:49.400
We said that total area is 1.
00:08:49.400 --> 00:08:59.300
It always has to increase and it always has to end up at 1.
00:08:59.300 --> 00:09:04.200
The cumulative distribution function always has the same general shape.
00:09:04.200 --> 00:09:16.200
It always starts at 0, at –infinity, and it always increases and it finishes up at 1, at infinity.
00:09:16.200 --> 00:09:22.100
That is why we have these properties F of infinity is 1, F is always increasing.
00:09:22.100 --> 00:09:30.700
Its derivative, this is the fundamental theorem of calculus, get you back to the density function which is f of Y.
00:09:30.700 --> 00:09:39.400
We will be using this property in particular, as we solve some problems.
00:09:39.400 --> 00:09:49.500
Let us jump in and solve some problems with density functions and cumulative distribution functions.
00:09:49.500 --> 00:09:58.300
The first one we are given that Y has density function f of Y, be careful, that is the distinction between f and F.
00:09:58.300 --> 00:10:02.400
C × Y, when Y is between 0 and 2.
00:10:02.400 --> 00:10:07.300
C × 4 – Y, when Y is between 2 and 4, and 0 elsewhere.
00:10:07.300 --> 00:10:11.200
I will draw a little graph of that.
00:10:11.200 --> 00:10:12.400
We do not know what C is.
00:10:12.400 --> 00:10:18.800
In fact, the first part of the problem here is to figure out what C should be.
00:10:18.800 --> 00:10:23.500
When Y goes from 0 to 2, we do know that it is something increasing.
00:10:23.500 --> 00:10:26.100
It is linear and it is increasing.
00:10:26.100 --> 00:10:30.100
4 - Y it will be decreasing again, as we go down to 4.
00:10:30.100 --> 00:10:34.900
It is something like that but we want to find the exact value of C.
00:10:34.900 --> 00:10:45.600
What we are going to use for that is, we are going to use the property of density functions which is that the integral of F of Y DY,
00:10:45.600 --> 00:10:50.300
the total area always has to be equal to 1.
00:10:50.300 --> 00:10:55.100
That was the first property of density function, I guess it was the second property of density functions.
00:10:55.100 --> 00:10:57.200
It always has to be equal 1.
00:10:57.200 --> 00:11:01.000
That represents the fact that this does represent probability,
00:11:01.000 --> 00:11:05.900
the total probability of something happening always has to be 1.
00:11:05.900 --> 00:11:10.000
Let us find that integral based on the information we are given.
00:11:10.000 --> 00:11:16.000
I'm only going really look at the range between 0 and 4.
00:11:16.000 --> 00:11:23.600
The integral from 0 to 4 of f of Y DY, I’m solving part A here.
00:11:23.600 --> 00:11:25.900
It is supposed to be equal to 1.
00:11:25.900 --> 00:11:30.900
Whatever that turns out to be, I'm going to set it equal to 1.
00:11:30.900 --> 00:11:34.700
We have got 2 different parts of our density function.
00:11:34.700 --> 00:11:39.900
I’m going to split up into integral from 0 to 2.
00:11:39.900 --> 00:11:44.400
We also have an integral from 2 to 4.
00:11:44.400 --> 00:11:51.900
Each one of those is multiplied by C, I will go ahead and factor the C out, put a C out here.
00:11:51.900 --> 00:12:00.300
Then, I will have Y DY here on this range and 4 - Y on this range.
00:12:00.300 --> 00:12:02.200
I'm just going to do the calculus there.
00:12:02.200 --> 00:12:20.200
This is C × Y²/2 evaluated from 0 to 2 + 4 – Y, that is 4Y - Y²/2, evaluate that from 2 to 4,
00:12:20.200 --> 00:12:22.500
that is supposed to be equal to 1.
00:12:22.500 --> 00:12:25.700
This is all C, always multiply by C.
00:12:25.700 --> 00:12:35.000
Y²/2 evaluated at 2 is 4/2 is just 2 evaluated at 0, nothing happens.
00:12:35.000 --> 00:12:43.200
If I plug in 4 into Y for the second part, I will get 16 – 4² that is 8.
00:12:43.200 --> 00:12:56.100
Plug in 2, I get -8, - and - is +, 2²/2 is + 4/2.
00:12:56.100 --> 00:12:58.200
That is still supposed to equal 1.
00:12:58.200 --> 00:13:03.200
I get C × 16 - 8 – 8, those cancel each other out.
00:13:03.200 --> 00:13:05.700
C × 4 = 1.
00:13:05.700 --> 00:13:14.300
C = ¼.
00:13:14.300 --> 00:13:19.500
Let us see, that tells me the answer to part A there.
00:13:19.500 --> 00:13:27.900
The key part there is that the total integral was supposed to come out to be 1, that is the property of a density function.
00:13:27.900 --> 00:13:32.000
The way I worked that out was, I set the total of the integral is equal to 1.
00:13:32.000 --> 00:13:42.900
I factored the C out of everything because that was a common factor on both parts there.
00:13:42.900 --> 00:13:46.400
Then I worked out this integral on their respective ranges.
00:13:46.400 --> 00:13:51.700
I had to split it up into 2 parts because the function was sort of defined piece wise.
00:13:51.700 --> 00:13:55.800
Two different definitions on 2 different ranges.
00:13:55.800 --> 00:14:01.700
We worked out the integral I got 2 × 4 is still equal to 1, it has to be equal to ¼.
00:14:01.700 --> 00:14:08.200
I got my part C, I have found my value of C for part A.
00:14:08.200 --> 00:14:13.200
Now, I have to find the cumulative distribution function of Y and that is going to be a little more work.
00:14:13.200 --> 00:14:20.300
Let me go ahead and do that on the next slide.
00:14:20.300 --> 00:14:29.500
With density function again, and the cumulative distribution function of Y, remember that F of Y,
00:14:29.500 --> 00:14:39.700
the way we figure that out is, it is the integral from -infinity to Y of F of T DT.
00:14:39.700 --> 00:14:44.700
That was our original definition of F, you can go back and you can find that.
00:14:44.700 --> 00:14:53.700
Now, in this case, there is nothing going on between -infinity and 0.
00:14:53.700 --> 00:15:00.400
I’m just going to start this integral at 0, the integral from 0 to Y.
00:15:00.400 --> 00:15:02.700
I have 2 different ranges of Y here.
00:15:02.700 --> 00:15:05.000
I’m going to split up 2 cases here.
00:15:05.000 --> 00:15:13.200
The first case I’m going to do is Y is between 0 and 2.
00:15:13.200 --> 00:15:14.200
What do I get there?
00:15:14.200 --> 00:15:23.100
Then, on the previous slide, we figured out that C is ¼.
00:15:23.100 --> 00:15:25.300
That is by the work we did on the previous slide.
00:15:25.300 --> 00:15:29.200
If you do not remember that, just go back and check it, you will see where we got the C is ¼.
00:15:29.200 --> 00:15:42.500
This is ¼, replacing my Y with T, I did the same mistake that I said that probability students often make, T DT.
00:15:42.500 --> 00:15:49.200
This is T²/2T²/2 × 4 is 2²/8.
00:15:49.200 --> 00:16:01.100
From 0 to Y which is just Y²/8, that is what my range of Y between 0 and 2.
00:16:01.100 --> 00:16:07.300
For the range between 2 and 4, it is more complicated and this is very easy for students to mix up.
00:16:07.300 --> 00:16:09.900
I hope you will follow me carefully here.
00:16:09.900 --> 00:16:17.400
F of Y, it is still the integral from 0 to Y of F of T DT.
00:16:17.400 --> 00:16:21.600
We have to split up into 2 parts because there is 2 different parts of this range.
00:16:21.600 --> 00:16:45.000
It is the integral from 0 to 2 of ¼ of T DT + the integral from 2 to Y of the different definition which is C, C is × 4 – T DT.
00:16:45.000 --> 00:16:48.900
I ‘m reading that off from this part of the definition here.
00:16:48.900 --> 00:16:53.300
It is a little more complicated but if we be careful with that, we can keep it straight.
00:16:53.300 --> 00:17:00.900
The integral of 1/4 of T DT, I can factor out a ¼ out of everything there.
00:17:00.900 --> 00:17:03.300
I think that will make my life a little simpler here.
00:17:03.300 --> 00:17:06.500
I factor the 1/4 out everything.
00:17:06.500 --> 00:17:20.500
The integral of T DT is T²/2, we will be dividing that from 0 to 2 +, now that 1/4 is gone, it is 4T - T²/2,
00:17:20.500 --> 00:17:27.900
evaluate that from T = 2 to T = Y.
00:17:27.900 --> 00:17:37.300
This is 1/4 × 2²/2 is just 2, + I will plug in Y for T .
00:17:37.300 --> 00:17:44.500
4Y - Y²/2, I will in 2 for T so -8.
00:17:44.500 --> 00:17:54.000
And - is +, + 2²/2 is just 2, this is ¼.
00:17:54.000 --> 00:18:06.800
Now what do I have here, I have 4Y - Y²/2 + 2 + 2 - 8 that is -4.
00:18:06.800 --> 00:18:18.100
If I distribute that 4, I get Y - Y²/8 -1.
00:18:18.100 --> 00:18:20.900
It is kind of a messy formula there but that is what we are stuck with.
00:18:20.900 --> 00:18:27.900
Now, I found two different function for F of Y, depending on the different ranges we are in.
00:18:27.900 --> 00:18:29.700
I’m going to summarize that.
00:18:29.700 --> 00:18:45.100
My F of Y is equal to Y²/8 for 0 is less than Y, less than or equal to 2.
00:18:45.100 --> 00:19:02.800
It is equal to Y - Y²/8 – 1 for 2 less than Y less than or equal to 4, that is my F of Y.
00:19:02.800 --> 00:19:06.300
I should also mention what it does on the outsides of those ranges.
00:19:06.300 --> 00:19:14.500
It will be 0/Y is less than 0 because, remember, the cumulative distribution function all starts out a 0
00:19:14.500 --> 00:19:19.700
and it always goes up to 1 for bigger values of Y.
00:19:19.700 --> 00:19:24.700
For Y greater than 4, it will be 1.
00:19:24.700 --> 00:19:30.200
It is worthwhile somtimes to check that the values match on the endpoints.
00:19:30.200 --> 00:19:34.700
If you plug in 0 to Y²/8, you do indeed get 0.
00:19:34.700 --> 00:19:54.900
If you plug in 4 to Y = 4, we will get to the second part of the function 4 - 4²/8 -1 will give us 4 -16/8 is 2 -1, gives us 1.
00:19:54.900 --> 00:20:00.700
That matches up of what we said Y should be, when we get bigger than 4.
00:20:00.700 --> 00:20:03.200
That checks my work there.
00:20:03.200 --> 00:20:13.800
Let me box this whole solution here because this is all part of our solution.
00:20:13.800 --> 00:20:18.400
Finally, we have found a cumulative distribution function Y there.
00:20:18.400 --> 00:20:20.400
Let me remind you of the steps there.
00:20:20.400 --> 00:20:29.200
We used the definition F of Y is the integral from -infinity to Y of F of T DT.
00:20:29.200 --> 00:20:34.700
Now, that is pretty simple when Y is between 0 and 2.
00:20:34.700 --> 00:20:41.300
You just take the first part of the definition and you work out the integral, and you get Y²/8.
00:20:41.300 --> 00:20:43.400
That is where we got this part of the answer.
00:20:43.400 --> 00:20:48.300
But when Y is between 2 and 4, it is much more complicated because you have
00:20:48.300 --> 00:20:52.400
to take into account both parts of this definition.
00:20:52.400 --> 00:20:58.100
You have to use both parts of this definition and split up the integral into two parts,
00:20:58.100 --> 00:21:03.600
and evaluate both of those using T as your variable.
00:21:03.600 --> 00:21:11.200
You do one from 0 to 2, one from 2 to Y, and then simplify that down into a much more complicated function.
00:21:11.200 --> 00:21:17.400
That is how we got this more complicated function on the range between 2 and 4.
00:21:17.400 --> 00:21:21.600
I want you to hang onto the answers that we got here for example 1,
00:21:21.600 --> 00:21:33.300
because we are going to use the same density function and therefore, the same cumulative distribution function for example 2.
00:21:33.300 --> 00:21:36.500
I want to make sure that you understand these answers for example 1.
00:21:36.500 --> 00:21:44.300
Make sure you understand this example very well, before you move on to example 2.
00:21:44.300 --> 00:21:49.100
In example 2 here, we are taking the same density function from example 1.
00:21:49.100 --> 00:21:52.200
Remember, we figured out that the constant had to be 1/4 there.
00:21:52.200 --> 00:21:55.400
I went ahead and write that in on example 2.
00:21:55.400 --> 00:22:05.300
It is ¼ Y for the range between 0 and 2, and 1/4 of 4 – Y from the range between 2 and 4.
00:22:05.300 --> 00:22:10.800
You got a density function that looks kind of like this.
00:22:10.800 --> 00:22:14.900
What we want to find here is the probability that Y is between 1 and 3,
00:22:14.900 --> 00:22:21.800
and the probability that Y is less than or equal to 2, given that Y is greater than or equal to 1.
00:22:21.800 --> 00:22:25.300
Those are some conditional probability involved in there.
00:22:25.300 --> 00:22:31.000
The useful thing to use at this point is not the density function that is given,
00:22:31.000 --> 00:22:36.100
but the cumulative distribution function that we worked out in example 1.
00:22:36.100 --> 00:22:41.600
If you have not just watched the video for example 1, you should go back and work out example 1
00:22:41.600 --> 00:22:47.800
because we are going to use that answer from example 1, to calculate the answers for example 2.2
00:22:47.800 --> 00:22:51.800
Let me remind you right now what the answer was from example 1.
00:22:51.800 --> 00:23:00.400
F of Y, the cumulative distribution function was, I broke it down into two important parts there.
00:23:00.400 --> 00:23:07.300
It was Y²/8, when Y is between 0 and 2.
00:23:07.300 --> 00:23:19.700
It was more complicated Y - Y²/8 -1, when 2 is less than Y less than 4.
00:23:19.700 --> 00:23:24.300
That was the cumulative distribution function, we did figure that out in example 1.
00:23:24.300 --> 00:23:28.200
Quite a lot of integration we went into that, we are not going to redo it.
00:23:28.200 --> 00:23:36.200
If you do not know where that is coming from, it is worth going back and working through example 1 because it will make sense.
00:23:36.200 --> 00:23:41.100
For part A here, to find the probability that Y is between 1 and 3.
00:23:41.100 --> 00:23:49.800
What we can do know is, we can use the cumulative distribution function F of 3 – F of 1.
00:23:49.800 --> 00:23:57.600
We can also use an integral of the density function but then, we just end up redoing the work we did from example 1.
00:23:57.600 --> 00:24:02.700
It is much easier to use F, if you already down that work.
00:24:02.700 --> 00:24:07.100
F of 3, now, 3 is between 2 and 4.
00:24:07.100 --> 00:24:12.800
Let me use this second version of the formula.
00:24:12.800 --> 00:24:20.200
It is 3 -, 3²/8 is 9/8 – 1.
00:24:20.200 --> 00:24:27.800
-F of 1, I have to use the first part of the formula because 1 is in the range between 0 and 2.
00:24:27.800 --> 00:24:34.000
Y square/8 is 1/8 and I will just simplify those fractions.
00:24:34.000 --> 00:24:40.200
3 -1 is 2 – 9/8 - 1/8 is 10/8.
00:24:40.200 --> 00:24:47.300
10/8 is 2 - 5/4 and 2 is 8/4, that is just ¾.
00:24:47.300 --> 00:24:51.700
That is my probability that Y is between 1 and 3.
00:24:51.700 --> 00:24:55.400
3/4 probability that Y is between 1 and 3.
00:24:55.400 --> 00:24:59.300
Let me show you that on the graph because I think it will makes sense there.
00:24:59.300 --> 00:25:05.300
There is 1, 2, 3, 4, being between 1 and 3.
00:25:05.300 --> 00:25:12.400
If you figure out how much of that area is between 1 and 3, if you do a little triangle geometry there,
00:25:12.400 --> 00:25:20.400
you will figure out that, that ¾ of the total area is between 1 and 3.
00:25:20.400 --> 00:25:26.200
We also checked that using our arithmetic here, using our integration.
00:25:26.200 --> 00:25:29.200
I’m going to jump over onto the next line to do part B.
00:25:29.200 --> 00:25:31.900
It will be a little more complicated, I need more space.
00:25:31.900 --> 00:25:34.700
Just remind you how we did part A here.
00:25:34.700 --> 00:25:40.700
I have recalled from example 1, the cumulative distribution function, the F of Y.
00:25:40.700 --> 00:25:42.700
That is what we worked out in example 1.
00:25:42.700 --> 00:25:49.000
And then, I just had to plug in F of 3 - F of 1.
00:25:49.000 --> 00:25:53.500
The wrinkle in that was those are two different ranges so I have to use two different formulas.
00:25:53.500 --> 00:25:59.500
One for F of 1, there is the F of 1 using that formula right there.
00:25:59.500 --> 00:26:07.600
There is F of 3, using that formula right there.
00:26:07.600 --> 00:26:13.300
Once I take those two values and plugged them in, I got some easy fractions to simplify.
00:26:13.300 --> 00:26:18.300
It ended up with ¾ there.
00:26:18.300 --> 00:26:19.600
We are still working on example 2.
00:26:19.600 --> 00:26:23.300
We still have to do the second part of the problem here.
00:26:23.300 --> 00:26:31.800
It is going to be really helpful to use the cumulative distribution function that we figured out in part 1.
00:26:31.800 --> 00:26:36.900
Let me remind you what our cumulative distribution function was.
00:26:36.900 --> 00:26:39.400
We figure this out in example 1.
00:26:39.400 --> 00:26:51.100
There was two parts to this function, Y²/8 for Y between 0 and 2.
00:26:51.100 --> 00:27:01.000
Y -1²/8 -1 for Y between 2 and 4.
00:27:01.000 --> 00:27:06.100
We got to find conditional probability, it has been a long time since we did condition probability.
00:27:06.100 --> 00:27:13.200
If you look back into some of the early videos in the series, you will find one that covers condition probability.
00:27:13.200 --> 00:27:16.400
I will remind you of the formula that we have for condition probability.
00:27:16.400 --> 00:27:30.900
The probability of A given B is the probability of A intersect B divided by the probability of B.
00:27:30.900 --> 00:27:36.800
Remember, intersection is like saying N, you want both of those things to be true.
00:27:36.800 --> 00:27:40.300
That formula is way back in the early videos for this course.
00:27:40.300 --> 00:27:46.300
You can find it, just scroll back to the videos here on www.educator.com.
00:27:46.300 --> 00:27:55.200
Our probability of Y being less than 2 given that it is greater than 1 is the probability of Y
00:27:55.200 --> 00:28:05.100
less than or equal to 2 and is greater than or equal to 1 divided by the probability that Y is greater than or equal to 1.
00:28:05.100 --> 00:28:08.100
I’m just filling in my formula for conditional probability.
00:28:08.100 --> 00:28:20.100
Now, another way of saying that it is less than 2 and greater than 1 is to say 1 is less than or equal Y less than or equal to 2 divided by,
00:28:20.100 --> 00:28:27.100
Now, the probability that Y is greater than or equal to 1, that is hard to compute directly but it is easy to compute
00:28:27.100 --> 00:28:32.600
if write is a 1- the probability that Y is less than or equal to 1.
00:28:32.600 --> 00:28:40.900
That is the easy way to calculate it because that sets it up to be something that we can answer
00:28:40.900 --> 00:28:44.500
using the cumulative distribution function.
00:28:44.500 --> 00:28:47.100
Let me write that on a new line here.
00:28:47.100 --> 00:28:53.700
The probability that Y is less than or equal to 1 is just F of 1.
00:28:53.700 --> 00:29:01.800
The probability that Y is between 1 and 2 is F of 2 – F of 1.
00:29:01.800 --> 00:29:09.100
Now, I can just input all these values into my cumulative distribution function, F.
00:29:09.100 --> 00:29:16.900
F of 2, it looks like everything here is on the first range, the Y²/8 which is nice, because that is the easier formula.
00:29:16.900 --> 00:29:22.700
F of 2 is 2²/8 that is 4/8.
00:29:22.700 --> 00:29:30.200
F of 1 is 1²/8, 1/8, 1 – F of 1 is 1 – 1/8.
00:29:30.200 --> 00:29:39.800
This is 3/8 /7/8, and if you do the flip on the fractions, the 8’s will cancel.
00:29:39.800 --> 00:29:44.000
I will just multiply top and bottom by 8, I will get 3/7.
00:29:44.000 --> 00:29:52.700
That is my probability that Y is less than or equal to 2 given that Y is greater than or equal to 1.
00:29:52.700 --> 00:29:57.300
You can also check that geometrically, if you like.
00:29:57.300 --> 00:30:05.300
The graph we have on f of Y looked like an elongated triangle, f of Y.
00:30:05.300 --> 00:30:11.100
There is 2, there is 1, and there is 3.
00:30:11.100 --> 00:30:20.300
The probability that Y is greater than or equal to 1 is all that range there.
00:30:20.300 --> 00:30:27.600
The probability that Y is less than or equal and 2, let me draw that in another color.
00:30:27.600 --> 00:30:32.700
Less than or equal to 2 is that range right there.
00:30:32.700 --> 00:30:43.700
If you break this up into little triangles, you can see that there is 1, 2, 3, 4, 5, 6, 7 triangles total.
00:30:43.700 --> 00:30:52.900
3 of these 7 little triangles are in that region.
00:30:52.900 --> 00:30:58.200
It does check graphically that we get this 3/7 answer.
00:30:58.200 --> 00:31:02.600
But I probably, do not want to rely on that, I do like using the formulas.
00:31:02.600 --> 00:31:05.300
Just to remind you how we did use the formulas there.
00:31:05.300 --> 00:31:11.300
This formula for the cumulative distribution function, this F came from example 1, we work that out in example 1.
00:31:11.300 --> 00:31:18.200
You can go back and check it, rewatch the video from example 1.
00:31:18.200 --> 00:31:22.600
This break down here, we are using conditional probability.
00:31:22.600 --> 00:31:27.900
This came from a very old video but it is on the series for conditional probability.
00:31:27.900 --> 00:31:33.200
I'm using that formula for conditional probability here.
00:31:33.200 --> 00:31:39.600
The probability that Y is less than or equal to 2 and greater than or equal 1, that just means it is between 1 and 2.
00:31:39.600 --> 00:31:44.200
The probability that Y is greater than or equal 1 is hard to evaluate directly.
00:31:44.200 --> 00:31:50.400
I flipped it around and that is because the probability that Y is less than or equal to 1
00:31:50.400 --> 00:31:57.800
is something we can answer easily using our cumulative distribution function, our F.
00:31:57.800 --> 00:32:02.900
This is F of 2 - F of 1, 1 – F of 1.
00:32:02.900 --> 00:32:10.000
I just dropped those values of Y into this F formula, since they are all in the first range there,
00:32:10.000 --> 00:32:18.700
and simplify the fractions down to 3/7.
00:32:18.700 --> 00:32:30.700
In examples 3, we are given a new density function for Y, F of Y, f of Y, f is the density function, is some C.
00:32:30.700 --> 00:32:36.600
It does not tell us what C is, I guess we have to figure that out, on the range between 0 and 1.
00:32:36.600 --> 00:32:41.500
2C is on the range between 1 and 2, and everywhere else it is going to be 0.
00:32:41.500 --> 00:32:46.300
To the first task here is to find what c should be.
00:32:46.300 --> 00:32:51.400
The second task is to find a cumulative distribution function of Y.
00:32:51.400 --> 00:32:55.000
Let me go ahead and graph this.
00:32:55.000 --> 00:33:04.200
We got between 0 and 1, the value is c.
00:33:04.200 --> 00:33:14.100
Between 1 and 2, it jumps up to 2c.
00:33:14.100 --> 00:33:20.600
That is our density function right there, it is 0 everywhere else.
00:33:20.600 --> 00:33:25.200
We have a density function that looks like that, it is a step function there.
00:33:25.200 --> 00:33:30.200
You can answer this question pretty easily graphically, if you know what you are doing.
00:33:30.200 --> 00:33:33.600
Let me go ahead and show you the arithmetic here.
00:33:33.600 --> 00:33:37.700
Just to make sure that it makes sense using either method.
00:33:37.700 --> 00:33:44.600
The key point here is that the total area under the density function is always equal to 1.
00:33:44.600 --> 00:33:59.000
In this case, the area is the integral from, in this case 0 to 2 of F of Y DY, should be equal to 1.
00:33:59.000 --> 00:34:03.900
I want to figure out what the value of c should be, to make that equal to 1.
00:34:03.900 --> 00:34:10.600
Let us evaluate that integral, that is the integral from, since we got the function defined differently
00:34:10.600 --> 00:34:16.500
on the 2 different ranges, I will write it as the interval from 0 to 1.
00:34:16.500 --> 00:34:21.400
We also have an integral from 1 to 2.
00:34:21.400 --> 00:34:23.500
It looks like they are both going to be multiplied by c.
00:34:23.500 --> 00:34:26.700
I will go ahead and factor that c out.
00:34:26.700 --> 00:34:31.100
That will leave me with 1 on the first range, 1 DY.
00:34:31.100 --> 00:34:39.100
2 on the second arrange, 2 DY there.
00:34:39.100 --> 00:34:44.200
That is supposed to come out to be equal to 1.
00:34:44.200 --> 00:35:00.900
Now, that is c × the integral of I DY is just Y from 0 to 1 + 2Y from 1 to 2 is equal to 1.
00:35:00.900 --> 00:35:08.500
That is C × evaluation of Y from 0 to 1 is just 1.
00:35:08.500 --> 00:35:16.100
2Y is just 2 × 2 -2 × 1, that is + 2 is equal to 1.
00:35:16.100 --> 00:35:25.100
It looks like c is going to be 1/3, in order to make this be a valid density functions.
00:35:25.100 --> 00:35:27.600
That tells us the answer to part A.
00:35:27.600 --> 00:35:32.900
That is really not surprising if you go back and look at your graph here.
00:35:32.900 --> 00:35:41.100
If you go back here, the area of that first block is definitely c, because height × width is I 2C.
00:35:41.100 --> 00:35:49.000
A of that second block is 2C, the total area is 3C which should be 1.
00:35:49.000 --> 00:35:58.100
Definitely, I want my C to be 1/3, in order for that to be a valid density function.
00:35:58.100 --> 00:36:04.400
I got my C to be 1/3, and I still need to find F of Y.
00:36:04.400 --> 00:36:11.700
That will be a little more work but let me recap the work here because that throws away the slide.
00:36:11.700 --> 00:36:17.500
I use the fact that the total area under the density function is always equal to 1.
00:36:17.500 --> 00:36:20.500
That is true for any density function, that is a requirement.
00:36:20.500 --> 00:36:27.200
It is essentially saying that the total probability in any experiment has to be 1.
00:36:27.200 --> 00:36:32.500
Because it was defined on two different ranges here, I just split up that integral into two parts.
00:36:32.500 --> 00:36:35.500
It is 0 to 1, and 1 to 2.
00:36:35.500 --> 00:36:45.500
Those were both very easy integrals, I factored out the C already.
00:36:45.500 --> 00:36:50.600
That gave me 3C was equal to 1, I figure out that C is equal to 1/3.
00:36:50.600 --> 00:36:56.800
I could have figured that out from the graph because I know that the total area should be 1.
00:36:56.800 --> 00:37:03.200
If I just look at the blocks there, I get 3C is equal to 1, C is equal to 1/3.
00:37:03.200 --> 00:37:11.700
We are going to hang onto this and make the jump to the next slide, where we will figure out part B here.
00:37:11.700 --> 00:37:18.600
For part B of examples 3, we figured out in part A that C was equal to 1/3.
00:37:18.600 --> 00:37:22.100
That is already done on the previous slide.
00:37:22.100 --> 00:37:32.600
But we have to now find F of Y, we want to find the cumulative distribution function.
00:37:32.600 --> 00:37:35.400
Let us remember the definition of F of Y.
00:37:35.400 --> 00:37:44.100
It is the integral from -infinity to Y of f of T DT.
00:37:44.100 --> 00:37:49.400
In this case, there is no density below 0.
00:37:49.400 --> 00:38:00.200
I can just cut this off at 0, this is the integral from 0 to Y of F of T DT.
00:38:00.200 --> 00:38:03.200
I need to separate two ranges here.
00:38:03.200 --> 00:38:11.900
I have a range from Y going from 0 to 1, and then I will have another range from Y going from 1 to 2.
00:38:11.900 --> 00:38:15.600
I need to separate this problem into two parts.
00:38:15.600 --> 00:38:16.900
The second part will be a little more difficult.
00:38:16.900 --> 00:38:24.900
The first part is pretty easy because then, I'm just looking at the definition of Y between 0 and 1.
00:38:24.900 --> 00:38:36.400
This is we get that from, 0 to Y is just 1/3, that was my value of C DT.
00:38:36.400 --> 00:38:48.100
That is 1/3 T evaluated from 0 to Y which is just 1/3Y, that part is fairly easy.
00:38:48.100 --> 00:38:53.600
The second part is more complicated, you want to be careful about that.
00:38:53.600 --> 00:38:59.800
On the second range, Y goes from, I said 0 to 2, and I should have said 1 to 2.
00:38:59.800 --> 00:39:10.400
You do not just want to use this formula from 1 to 2 because we really need to find the integral from 0 to Y of F of T DT.
00:39:10.400 --> 00:39:24.000
That breaks that up into two parts, the integral from 0 to 1 of F of T DT and the integral from 1 to Y of F of T DT.
00:39:24.000 --> 00:39:27.100
We need to do separate integrals for each of those.
00:39:27.100 --> 00:39:30.600
I have to do that because I know that Y is bigger than 1.
00:39:30.600 --> 00:39:34.200
Y is somewhere in this range between 1 and 2.
00:39:34.200 --> 00:39:55.200
The first part is not too bad, 0 to 1 of 1/3 DT +, 1 to Y between 1 and 2, the density function is 2 × C, that is 2/3 DT.
00:39:55.200 --> 00:40:10.400
I get 1/3 T from the 0 to 1 + 2/3 T from 1 to Y, from T = 1 to T = Y.
00:40:10.400 --> 00:40:21.300
I get 1/3 × one – 0 so 1/3 × 1 + 2/3 Y - 2/3 × 1.
00:40:21.300 --> 00:40:31.300
I guess that simplifies a bits into 2/3 Y + 1/3 - 2/3 is -1/3.
00:40:31.300 --> 00:40:35.800
I got two different ranges in two different functions.
00:40:35.800 --> 00:40:52.500
I will put those together to give my answer, my F of Y is equal to 1/3Y, when 0 is less than Y less than 1.
00:40:52.500 --> 00:41:07.100
It is equal to 2/3 Y - 1/3, when one is less than or equal to Y less than or equal to 2.
00:41:07.100 --> 00:41:10.900
Those are the important parts of my cumulative distribution function.
00:41:10.900 --> 00:41:13.400
We just tacked on the parts of the N.
00:41:13.400 --> 00:41:23.900
If Y is less than 0, it will just be 0 because the left hand side is always 0, the right hand side is always 1.
00:41:23.900 --> 00:41:31.200
If Y is greater than 2 then we will be looking at a function of 1.
00:41:31.200 --> 00:41:38.000
You can always plug that in and check, I will plug in Y = 2 and I get 2/3 × 2.
00:41:38.000 --> 00:41:42.800
4/3 - 1/3, I’m plugging in Y = 2 in here and it does come out to be 1.
00:41:42.800 --> 00:41:51.400
That reassures me that I have probably been doing my arithmetic correctly.
00:41:51.400 --> 00:41:58.300
Let me box this up.
00:41:58.300 --> 00:42:00.900
I got my cumulative distribution function.
00:42:00.900 --> 00:42:04.300
We will be using this again in the next examples.
00:42:04.300 --> 00:42:09.900
I want to make sure that you understand this very well, before you move on to example 4.
00:42:09.900 --> 00:42:11.900
Let me remind you of how we got that.
00:42:11.900 --> 00:42:18.800
Our cumulative distribution function, the F, we always get that by integrating the f, the density function.
00:42:18.800 --> 00:42:25.200
You integrate it, you change the variable from Y to T, and then you integrate from 0 to Y.
00:42:25.200 --> 00:42:30.500
That is sometimes a bit subtle, especially, when you have these functions that are defined piece wise.
00:42:30.500 --> 00:42:41.400
Because the integral from 0 to Y, Y is between 0 and 1, no problem, you just use the definition of the f of Y between 0 and 1.
00:42:41.400 --> 00:42:43.800
You integrate that and you get 1/3Y.
00:42:43.800 --> 00:42:50.500
If Y is between 1 and 2, then you get a more complicated integral because
00:42:50.500 --> 00:42:54.500
you have to break it up between 0 to 1, and 1 to Y.
00:42:54.500 --> 00:43:02.700
And then, you have to use the different definitions, the 1/3 the 1C, the 2/3 the 2C, on the different ranges.
00:43:02.700 --> 00:43:13.000
You plug in those two different definitions, the C and the 2 C, and then you do the integral and it simplifies fairly nicely.
00:43:13.000 --> 00:43:20.100
You want to assemble these two answers into a definition of F of Y defined on the different ranges.
00:43:20.100 --> 00:43:25.900
You always have the left hand then being 0 and the right hand then being 1.
00:43:25.900 --> 00:43:28.800
That is our cumulative distribution function.
00:43:28.800 --> 00:43:33.500
You do want to remember this because we will be using it again for example 4.
00:43:33.500 --> 00:43:36.400
Let us go ahead and take a look at that.
00:43:36.400 --> 00:43:41.500
In example 4, we are using the same density function from example 3.
00:43:41.500 --> 00:43:50.200
F of Y is 1/3 on the range from 0 to 1, 2/3 on the range from 1 to 2.
00:43:50.200 --> 00:44:00.200
Here, we want to find a condition probability, the probability that Y is less than 3/2 given that it is bigger than ½.
00:44:00.200 --> 00:44:05.100
I did do a lot of work on this density function back in examples 3.
00:44:05.100 --> 00:44:11.800
If you did not just watch the video on example 3, you should go back and work through example 3,
00:44:11.800 --> 00:44:14.400
in order to understand example 4.
00:44:14.400 --> 00:44:17.000
I’m going to use the results right now.
00:44:17.000 --> 00:44:25.800
Let me remind you what our answer was from example 3.
00:44:25.800 --> 00:44:28.500
Let us see, where do I have space for that, I will put that over here.
00:44:28.500 --> 00:44:35.800
In example 3, this was our answer from examples 3 and it did take us a fair amount of work to get to this point.
00:44:35.800 --> 00:44:43.800
We found the cumulative distribution function F of Y not f, you got to be very careful to keep those clear.
00:44:43.800 --> 00:44:47.900
I will just put the two important ranges here.
00:44:47.900 --> 00:44:56.400
The range from 0 to 1, and then we had a different answer when Y was between 1 and 2.
00:44:56.400 --> 00:45:02.900
Our two answers there, where Y was 3, where do I have that in my notes there.
00:45:02.900 --> 00:45:12.800
That was 1/3 Y on the first range, and then on the second range it was 2/3 Y -1/3.
00:45:12.800 --> 00:45:17.200
We work those out in example 3, I’m not the showing you the work right now to do those.
00:45:17.200 --> 00:45:20.100
You have to go back and watch example 3.
00:45:20.100 --> 00:45:26.400
For the probability of what we are asking for here, this is the conditional probability,
00:45:26.400 --> 00:45:29.500
let me remind you of the formula for conditional probability.
00:45:29.500 --> 00:45:34.900
That was done in one of our earlier probability videos here on www.educator.com.
00:45:34.900 --> 00:45:40.900
Let me remind you that the probability of A given B, the conditional probability is
00:45:40.900 --> 00:45:49.900
the probability of A intersect B divided by the probability of B.
00:45:49.900 --> 00:46:00.100
Remember, A intersect B is the same as A and B, mathematical, when events are happening at the same time,
00:46:00.100 --> 00:46:02.200
divided by the probability of B.
00:46:02.200 --> 00:46:07.000
That is what we are going to apply to this conditional probability that we are asked to calculate here.
00:46:07.000 --> 00:46:20.000
This is the probability that Y is less than or equal to 3/2 and Y is greater than or equal to ½ divided by the probability of B.
00:46:20.000 --> 00:46:23.100
This is our event A and this is B right here.
00:46:23.100 --> 00:46:30.700
B is the probability that Y is greater than or equal to ½.
00:46:30.700 --> 00:46:42.900
Being less than 3/2 and bigger than ½, that just means you are in the range between ½ and 3/2.
00:46:42.900 --> 00:46:57.400
Being greater than ½, the easier way to think about that is to write that as 1 - the probability that Y is less than ½.
00:46:57.400 --> 00:47:03.000
The reason I put it that way is because that is making it in a form where
00:47:03.000 --> 00:47:06.500
we can easily use our cumulative distribution function.
00:47:06.500 --> 00:47:20.700
Remember, F of Y represents by definition, it represents the probability that Y is less than or equal to some value y.
00:47:20.700 --> 00:47:31.400
This is equal to, in the denominator that would be 1 – F of ½.
00:47:31.400 --> 00:47:45.000
In the numerator, we want the probability of being between ½ and 3/2, that is F of 3/2 – F of ½.
00:47:45.000 --> 00:47:53.900
Now that I have an F function, I work that out in example 3, I can just drop in all the values.
00:47:53.900 --> 00:48:01.000
3/2 is bigger than 1, I have to use the second range there F of 3/2.
00:48:01.000 --> 00:48:21.300
2/3 × 3/2 - 1/3 -1/2, now that ½ is less than 1, I will use the first range - 1/3 × ½ all divided by 1 - F of ½ is just 1/3 × ½.
00:48:21.300 --> 00:48:23.400
Now, I just have some fractions to simplify.
00:48:23.400 --> 00:48:35.400
2/3 and 3/2 that is 1 -1/3 - 1/3 × ½ is 1/6, 1 -1/6.
00:48:35.400 --> 00:48:42.000
1/3 + 1/6 is ½, I get 1 -1/2 is just ½ in the numerator.
00:48:42.000 --> 00:48:58.100
1 -1/6 is 5/6 in the denominator, that is 6/5 × ½ is 3/5 as my answer.
00:48:58.100 --> 00:49:02.100
I think this is one we can also check graphically.
00:49:02.100 --> 00:49:03.600
Let me draw that function.
00:49:03.600 --> 00:49:14.100
We graph this out in examples 3 but that was the density function that I'm graphing right now.
00:49:14.100 --> 00:49:18.500
It had a big jump at 1 and it really went from 0 to 2.
00:49:18.500 --> 00:49:24.100
The probability that you are less than 3/2, let me draw 3/2 there.
00:49:24.100 --> 00:49:29.900
There is 3/2, given that you are bigger than ½.
00:49:29.900 --> 00:49:40.600
There is ½ and ½, we are given that we are bigger than ½, let me go ahead and color in that area there.
00:49:40.600 --> 00:49:51.500
There is the probability that we are bigger than ½ and being less than 3/2 would mean that we are in that area right there.
00:49:51.500 --> 00:49:53.700
I’m going to cut off at 3/2.
00:49:53.700 --> 00:50:03.900
If you just kind of work by blocks there, there is sort of 5 blocks in the color blue1, 2, 3, 4, 5,
00:50:03.900 --> 00:50:06.800
and three of them are colored green.
00:50:06.800 --> 00:50:14.600
That is where we get the 3/5 coming from, if you want to do it graphically, instead of doing the equations.
00:50:14.600 --> 00:50:18.000
You will still get 3/5.
00:50:18.000 --> 00:50:21.100
Let me recap how we did that.
00:50:21.100 --> 00:50:24.000
We are using conditional probability here.
00:50:24.000 --> 00:50:27.100
I went back to my old conditional probability formula.
00:50:27.100 --> 00:50:32.500
The probability of A and B divided by the probability of B.
00:50:32.500 --> 00:50:38.100
In this case, my A and B where Y being less than 3/2 and bigger than ½.
00:50:38.100 --> 00:50:42.300
I get the probability that Y is between ½ and 3/2.
00:50:42.300 --> 00:50:50.100
The probability of Y being bigger than ½, I flip that around and said it is 1 - the probability of Y being less than ½.
00:50:50.100 --> 00:50:59.900
The reason I flipped it around was, I could easily convert that into a value of my cumulative distribution function, my F.
00:50:59.900 --> 00:51:04.100
I can also find this probability of the range using F.
00:51:04.100 --> 00:51:12.500
I recalled the F that we calculated back in examples 3.
00:51:12.500 --> 00:51:18.600
I recalled this F here and I just dropped in a different values.
00:51:18.600 --> 00:51:28.100
Of course, I have to use the different parts of the formula because 3/2 was in this second range and ½ was in this first range.
00:51:28.100 --> 00:51:36.100
That is why I use the second formula for 3/2 and the first formula for ½.
00:51:36.100 --> 00:51:40.200
Then, I just simplify down the fractions and it simplify down to 3/5.
00:51:40.200 --> 00:51:46.200
I could have done all that just by looking at the graph and by measuring up the areas.
00:51:46.200 --> 00:51:56.000
I get 3 blocks out of 5 blocks, that is why that checks my answer to be 3/5.
00:51:56.000 --> 00:52:03.500
In example 5 here, we have a cumulative distribution function given to us, the F of Y.
00:52:03.500 --> 00:52:10.200
It looks like the most important ranges here are Y being between 0 and 1, and Y being between 1 and 2.
00:52:10.200 --> 00:52:15.300
We got two problems here, we want to find f of Y, the density function.
00:52:15.300 --> 00:52:22.700
We also want to find the probability that Y is in a particular range.
00:52:22.700 --> 00:52:26.000
It looks like I have really left myself no space to solve this problem.
00:52:26.000 --> 00:52:34.500
Let me jump over to the next slide and solve the problem.
00:52:34.500 --> 00:52:44.100
Here is my cumulative distribution function, I put off the less important parts of the definition there.
00:52:44.100 --> 00:52:50.500
We are going to find f of Y and the probability that Y is in a certain range.
00:52:50.500 --> 00:52:54.800
This is fairly easy relative to the earlier problems that we did.
00:52:54.800 --> 00:52:57.200
You get a little bit of break this time.
00:52:57.200 --> 00:53:04.900
What is this X doing here, we do not use X in probability.
00:53:04.900 --> 00:53:11.100
F of Y is equal to, remember, it is the derivative of F of Y.
00:53:11.100 --> 00:53:15.900
I can find this just by taking the derivative of F of Y.
00:53:15.900 --> 00:53:25.700
The derivative of Y/4 is ¼ and the derivative of Y²/4 is 2Y/4, that is just Y/2.
00:53:25.700 --> 00:53:36.400
That is for the particular ranges 0 less than Y less than or equal to 1, and 1 less than Y less than or equal 2.
00:53:36.400 --> 00:53:42.700
Outside those ranges, F of Y, remember it was just the constant 0 and 1.
00:53:42.700 --> 00:53:50.200
Its derivative will just be 0 for other values of y.
00:53:50.200 --> 00:53:53.800
If Y is less that or equal to 0, f will be 0.
00:53:53.800 --> 00:54:02.500
If Y is greater than 2, f is greater than 2, f will also be 0.
00:54:02.500 --> 00:54:10.000
That is my answer for f of Y, we found the density function there by taking
00:54:10.000 --> 00:54:17.100
the derivative of the cumulative distribution function.
00:54:17.100 --> 00:54:19.500
Those are the answer to part A.
00:54:19.500 --> 00:54:27.500
Part B, we want to find the probability that Y is between ½ and 3/2.
00:54:27.500 --> 00:54:36.900
You can do that quickly just by using the cumulative distribution function F of 3/2 - F of ½.
00:54:36.900 --> 00:54:42.200
You do not have to do any integral here because we already have the cumulative distribution function.
00:54:42.200 --> 00:54:43.000
Let us just drop those in.
00:54:43.000 --> 00:54:53.700
Now, 3/2 is between 1 and 2, I’m going to use the second formula there, 3/2²/4.
00:54:53.700 --> 00:55:01.100
For ½, I use the first part of the definition that is because ½ is between 0 and 1.
00:55:01.100 --> 00:55:08.200
½ /4, there should be a - there not =.
00:55:08.200 --> 00:55:15.300
I just work out the fractions, that is 9/4 /4 - ½ /4.
00:55:15.300 --> 00:55:26.600
½ /4 is 9/16 - 1/8, we can write 1/8 as 2/16.
00:55:26.600 --> 00:55:34.900
9/16 - 2/16 is 7/16, that is my answer.
00:55:34.900 --> 00:55:40.700
You do not have to do any integrals there, and that is because the cumulative distribution function was already given to us.
00:55:40.700 --> 00:55:45.300
If it had been a density function, we would have been doing the integral to calculate that.
00:55:45.300 --> 00:55:47.700
Let me recap the steps there.
00:55:47.700 --> 00:55:53.900
We are given in this one, we are given the cumulative distribution function and we are trying to find the density functions.
00:55:53.900 --> 00:55:57.100
We are given F, we are trying to find f.
00:55:57.100 --> 00:56:00.400
To find f, what you will just do is, go and take the derivative.
00:56:00.400 --> 00:56:06.400
You took the derivative of Y/4, that is where that 1/4 came from.
00:56:06.400 --> 00:56:13.400
Took the derivative of Y²/4, that is where that Y/2 came from because it is 2Y/4.
00:56:13.400 --> 00:56:17.900
The derivatives of the constants on either end are just 0.
00:56:17.900 --> 00:56:22.100
We get those two definitions on those ranges.
00:56:22.100 --> 00:56:24.100
That is how we find f of Y.
00:56:24.100 --> 00:56:31.400
To find probabilities, if you know the cumulative distribution function, it is just a matter of plugging the endpoints into F.
00:56:31.400 --> 00:56:38.300
The only issue there is you have to be careful which of the two definitions for F you use.
00:56:38.300 --> 00:56:43.400
You figured that out because 3/2 is between 1 and 2.
00:56:43.400 --> 00:56:46.300
1/2 is between 0 and 1.
00:56:46.300 --> 00:56:51.700
That is why I used those two respected definitions for F.
00:56:51.700 --> 00:56:58.600
Drop the numbers in, simplify it down, get a nice fraction as my answer.
00:56:58.600 --> 00:57:04.200
That wraps up our lecture on density functions and cumulative distribution functions.
00:57:04.200 --> 00:57:08.600
This is part of the chapter on continuous probability,
00:57:08.600 --> 00:57:14.600
which in turn is part of the probability lecture series here on www.educator.com.
00:57:14.600 --> 00:57:17.000
My name is Will Murray, thank you very much for joining us, bye.