Probability Theory - Part 10 - Random Variables

0:19
We just want to put all the relevant information of a random experiment into one object.
0:25
and usually, for such objects, we use capital letters. For example capital X.
0:31
Now soon we will see that this capital X is just a map defined on a sample space Omega with some special properties.
0:39
Indeed often here the codomain is simply given by the real number lin R.
0:45
However in a moment i will show you how we can naturally generalize this to another set of values.
0:51
and of course we will also discuss the properties we need for such a map.

 

0:56
But before we do this lets first discuss a simple example.
1:01
So what we could do is, as often, throwing 2 dice and maybe we have a red one and a green one.
1:07
Hence we can distinguish the dice, which means this is the same as throwing one die twice.
1:14
Therefore you should immediately be able to write down the probability space, because this is exactly what we discussed in the last video.
1:22
So the sample space Omega is the cartesian product of 1 to 6 with itself and the sigma-algebra A is just the power set.
1:30
and then without writing down the explicit definition, we can simply say, P is given by the uniform distribution.
1:37
Ok there you see, with this probability space we have the whole information of this random experiment.
1:44
So all the possible outcomes with the probabilities are given here(this probability space).

 

1:49
However, maybe we are in a situation where we are only interested in the sum of the two numbers the dice show.
1:56
For example, we could be in a game where this is important and the colours of the two dice don't matter at all.
2:03
Then exactly in such a case, we would define a random variable.
2:08
and as I told you before, we would simply call it X.

 

2:12
Ok, so here we have a map from the sample space Omega, which has all the possible outcomes of the 2 dice, to the real numbers.
2:21
So what X should do we already know. It should give us the sum of the 2 numbers.
2:26
Hence a sample given by (omega_1, omega_2) is mapped to the sum omega_1 + omega_2.
2:34
So you see, this is not a complicated map at all.
2:38
What we can remember in this case is that the input is a sample and the output is a number.
2:44
Ok, so this is a typical example of a random variable, where you see it simply extracts the information we are interested in.

2:53
Therefore later we will work with a lot of random variables.
2:57
However, first, I would say we have to give the correct definition now.
3:02
and as promised, this is the general definition one uses in probability theory.

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3:07
Ok, so what we need are 2 spaces we call measurable spaces or event spaces.
3:13
The first one should be given by a set. A sample space Omega with a corresponding sigma algebra A
3:20
and then the second one is given by a set Omega tilde with a corresponding sigma algebra A tilde.
3:26
So you could say we have here probability spaces, where the probability measure P is not fixed yet.
3:33
Hence we are just interested in the events, the elements of a sigma algebra and therefore we talk of events spaces.
3:40
However, in measure theory, we would call these spaces measurable spaces
3:46
and in fact the random variable we define now, we would call a measurable map.
3:51
Nevertheless as often in probability theory, we use some special names for these objects.
3:57
Ok, now we consider a map we call capital X from one sample space Omega into the other one, Omega tilde
4:05
and then this map is called a random variable if it is a measurable map in the measure theoretical sense.

4:12
Which means we have to look at all the pre-images of the events in the second sample space, Omega tilde.

 

4:18
Here let's denote an element of curved A tilde, just with a normal A tilde.
4:24
and then we know, this pre-image of A tilde is the subset of Omega.
4:29

However, in the end, when we have a probability measure P, we want to measure these sets here.

4:36
Hence it's necessary that this is not just a subset of Omega, but also an element of the sigma-algebra A.
4:43

Therefore this is exactly the right condition we need here for all A tilde.

4:49
OK, there we have it. This is the whole definition of a random variable.
4:54
A concept we will need a lot.

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4:56
Therefore I would say let's immediately look at some examples.
5:00
So maybe for the start lets discuss the details of the random variable from above.
5:05
There the first event space was given by (Omega, A).
5:09
Where Omega is the sample space given by 1 to 6, squared and A is just the power set.
5:15
Moreover the second event space was given by the real number line.
5:20
Hence this would be Omega tilde.
5:24
However, now we should ask: What is A tilde?
5:28
In the end, i can already tell you it will not matter at all,
5:31
but usually, when we have the real number line, we would take the Borel sigma-algebra.
5:37
Therefore we also do this here
5:40
and now I can ask you: is this map from before, this X, actually a random variable?
5:47
At first glance, it seems to be that we have to check a lot, because we need to check all the pre-images here.
5:54
However please recall that the sigma-algebra A here is the whole power set.
5:59
Hence the condition we have to satisfy just tells us that the pre-image of any Borel set A tilde is a subset of Omega.
6:09
Which is of course trivially fulfilled.
6:12
This means that the definition of X does not matter at all, when we have on the left-hand side the whole power set as the sigma-algebra.
6:20
So in this case, we can easily conclude that the map X is a random variable.

 

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6:25
Indeed most of the time we won't have any problems at all fulfilling this condition here
6:30
or to put it in other words, counterexamples are always very artificial.
6:35
For example, you could take the same case again, but now we will change the power set here. So we take another sigma algebra A.
6:44
Hence instead of the largest one, the power set, we take the smallest one.
6:49
So the only events we have in our sigma algebra A are the empty set and Omega itself.
6:55
When of course we immediately find a counter-example. You just have to look at the pre-image of the singleton 2.
7:01
In words, this means the sum of the 2 numbers of the dice is exactly 2.
7:07
So there is only one dice throw possible. The 2 dice show both one.
7:12
Therefore this pre-image is just a set with only one element.
7:17
and that's the reason I chose 2 here, because then I don't have to write so much
7:21
and of course, you immediately see the result.
7:24
This set is neither the empty set nor the whole set Omega.
7:29
So it's not an element in the sigma-algebra A.
7:32
and then we can conclude, in this case, X is not a random variable.

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7:37
Ok, now in summary what you should see here is: random variables are not complicated at all.
7:43
and indeed most of the time the fact that we have a random variable is immediately given.
7:49
Ok, now I want to close this video with an important notation.
7:53
Assume again, that we have 2 measurable spaces also called event spaces.
7:59
Moreover, let's also fix two other things.
8:02
First, we have a random variable X as before
8:05
and secondly maybe not so surprising, we have a probability measure P.
8:09
Defined on the first event space on the left.
8:13
This means when we take any set A tilde from the sigma-algebra A tilde
8:18
and look at the pre-image under X, then we can calculate the probability of this event.
8:24
Simply because we know by the definition of a random variable that this set lies in the sigma-algebra A.
8:32

Hence P of this set makes sense.

 

 

 

8:35
Therefore one usually uses a shorter, but strange notation for this.
8:40
One simply writes P of X in A tilde.
8:45
First it looks a little bit odd, but you will see this a lot in probability theory
8:51
and indeed it makes a little bit more sense when we use the definition of a pre-image.
8:56
Which is simply the set of all lower case omega in capital Omega
9:01
with the property that X of lower case omega lies in A tilde.
9:06
Hence you can see the left-hand side as a shortcut for writing this whole set.
9:11
The important thing you should remember here is that this literally does not make sense, but it stands for a whole set in Omega.
9:19
Please note in the same sense also other shortcuts as this are used as well.
9:24
In more detail, we will discuss this in the next video.
9:28
Therefore I hope I see there and have a nice day. Bye!