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Probability Theory - Part 10 - Random VariablesProbability Theory 2022. 9. 21. 05:18
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We just want to put all the relevant information of a random experiment into one object.
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and usually, for such objects, we use capital letters. For example capital X.
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Now soon we will see that this capital X is just a map defined on a sample space Omega with some special properties.
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Indeed often here the codomain is simply given by the real number lin R.
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However in a moment i will show you how we can naturally generalize this to another set of values.
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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.
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So what we could do is, as often, throwing 2 dice and maybe we have a red one and a green one.
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Hence we can distinguish the dice, which means this is the same as throwing one die twice.
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Therefore you should immediately be able to write down the probability space, because this is exactly what we discussed in the last video.
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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.
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and then without writing down the explicit definition, we can simply say, P is given by the uniform distribution.
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Ok there you see, with this probability space we have the whole information of this random experiment.
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So all the possible outcomes with the probabilities are given here(this probability space).
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However, maybe we are in a situation where we are only interested in the sum of the two numbers the dice show.
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For example, we could be in a game where this is important and the colours of the two dice don't matter at all.
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Then exactly in such a case, we would define a random variable.
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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.
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So what X should do we already know. It should give us the sum of the 2 numbers.
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Hence a sample given by (omega_1, omega_2) is mapped to the sum omega_1 + omega_2.
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So you see, this is not a complicated map at all.
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What we can remember in this case is that the input is a sample and the output is a number.
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Ok, so this is a typical example of a random variable, where you see it simply extracts the information we are interested in.
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Therefore later we will work with a lot of random variables.
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However, first, I would say we have to give the correct definition now.
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and as promised, this is the general definition one uses in probability theory.
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Ok, so what we need are 2 spaces we call measurable spaces or event spaces.
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The first one should be given by a set. A sample space Omega with a corresponding sigma algebra A
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and then the second one is given by a set Omega tilde with a corresponding sigma algebra A tilde.
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So you could say we have here probability spaces, where the probability measure P is not fixed yet.
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Hence we are just interested in the events, the elements of a sigma algebra and therefore we talk of events spaces.
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However, in measure theory, we would call these spaces measurable spaces
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and in fact the random variable we define now, we would call a measurable map.
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Nevertheless as often in probability theory, we use some special names for these objects.
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Ok, now we consider a map we call capital X from one sample space Omega into the other one, Omega tilde
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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.
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Here let's denote an element of curved A tilde, just with a normal A tilde.
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and then we know, this pre-image of A tilde is the subset of Omega.
4:29However, in the end, when we have a probability measure P, we want to measure these sets here.
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Hence it's necessary that this is not just a subset of Omega, but also an element of the sigma-algebra A.
4:43Therefore this is exactly the right condition we need here for all A tilde.
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OK, there we have it. This is the whole definition of a random variable.
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A concept we will need a lot.
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Therefore I would say let's immediately look at some examples.
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So maybe for the start lets discuss the details of the random variable from above.
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There the first event space was given by (Omega, A).
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Where Omega is the sample space given by 1 to 6, squared and A is just the power set.
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Moreover the second event space was given by the real number line.
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Hence this would be Omega tilde.
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However, now we should ask: What is A tilde?
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In the end, i can already tell you it will not matter at all,
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but usually, when we have the real number line, we would take the Borel sigma-algebra.
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Therefore we also do this here
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and now I can ask you: is this map from before, this X, actually a random variable?
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At first glance, it seems to be that we have to check a lot, because we need to check all the pre-images here.
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However please recall that the sigma-algebra A here is the whole power set.
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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.
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Which is of course trivially fulfilled.
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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.
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So in this case, we can easily conclude that the map X is a random variable.
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Indeed most of the time we won't have any problems at all fulfilling this condition here
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or to put it in other words, counterexamples are always very artificial.
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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.
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Hence instead of the largest one, the power set, we take the smallest one.
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So the only events we have in our sigma algebra A are the empty set and Omega itself.
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When of course we immediately find a counter-example. You just have to look at the pre-image of the singleton 2.
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In words, this means the sum of the 2 numbers of the dice is exactly 2.
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So there is only one dice throw possible. The 2 dice show both one.
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Therefore this pre-image is just a set with only one element.
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and that's the reason I chose 2 here, because then I don't have to write so much
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and of course, you immediately see the result.
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This set is neither the empty set nor the whole set Omega.
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So it's not an element in the sigma-algebra A.
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and then we can conclude, in this case, X is not a random variable.
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Ok, now in summary what you should see here is: random variables are not complicated at all.
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and indeed most of the time the fact that we have a random variable is immediately given.
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Ok, now I want to close this video with an important notation.
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Assume again, that we have 2 measurable spaces also called event spaces.
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Moreover, let's also fix two other things.
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First, we have a random variable X as before
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and secondly maybe not so surprising, we have a probability measure P.
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Defined on the first event space on the left.
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This means when we take any set A tilde from the sigma-algebra A tilde
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and look at the pre-image under X, then we can calculate the probability of this event.
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Simply because we know by the definition of a random variable that this set lies in the sigma-algebra A.
8:32Hence P of this set makes sense.
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Therefore one usually uses a shorter, but strange notation for this.
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One simply writes P of X in A tilde.
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First it looks a little bit odd, but you will see this a lot in probability theory
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and indeed it makes a little bit more sense when we use the definition of a pre-image.
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Which is simply the set of all lower case omega in capital Omega
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with the property that X of lower case omega lies in A tilde.
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Hence you can see the left-hand side as a shortcut for writing this whole set.
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The important thing you should remember here is that this literally does not make sense, but it stands for a whole set in Omega.
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Please note in the same sense also other shortcuts as this are used as well.
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In more detail, we will discuss this in the next video.
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Therefore I hope I see there and have a nice day. Bye!'Probability Theory' 카테고리의 다른 글
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