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Title: Random Variables
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Series: Probability Theory
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YouTube-Title: Probability Theory 10 | Random Variables
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Bright video: https://youtu.be/WPk-YahKyoE
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Dark video: https://youtu.be/2_t7UkbStuM
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Ad-free video: Watch Vimeo video
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Exercise Download PDF sheets
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: pt10_sub_eng.srt
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Timestamps
00:00 Intro/ short introduction
00:56 Example (discrete)
02:57 Definition of a random variable
04:56 Continuation of the example
07:49 Notation
09:28 Outro
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Subtitle in English
1 00:00:00,400 –> 00:00:03,694 Hello and welcome back to probability theory.
2 00:00:04,243 –> 00:00:09,247 and as always first i want to thank all the nice people that support this channel on Steady or Paypal.
3 00:00:09,657 –> 00:00:14,013 Now, in todays part 10 we will talk about so called random variables.
4 00:00:14,443 –> 00:00:19,114 First they might sound complicated, but actually they are very simple and natural.
5 00:00:19,857 –> 00:00:25,286 We just want to put all the relevant information of a random experiment into one object.
6 00:00:25,800 –> 00:00:31,080 and usually for such objects we use capital letters. For example capital X.
7 00:00:31,600 –> 00:00:39,273 Now soon we will see that this capital X is just a map defined on a sample space Omega with some special properties.
8 00:00:39,473 –> 00:00:44,808 Indeed often here the codomain is simply given by the real number lin R.
9 00:00:45,600 –> 00:00:51,168 However in a moment i will show you how we can naturally generalize this to another set of values.
10 00:00:51,614 –> 00:00:56,143 and of course we will also discuss the properties we need for such a map.
11 00:00:56,343 –> 00:01:00,249 But before we do this lets first discuss a simple example.
12 00:01:01,343 –> 00:01:07,478 So what we could do is, as often, throwing 2 dice and maybe we have a red one and a green one.
13 00:01:07,914 –> 00:01:13,863 Hence we can distinguish the dice, which means this is the same as throwing one die twice.
14 00:01:14,414 –> 00:01:22,058 Therefore you should immediately be able to write down the probability space, because this is exactly what we discussed in the last video.
15 00:01:22,258 –> 00:01:30,249 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.
16 00:01:30,449 –> 00:01:37,379 and then without writing down the explicit definition, we can simply say, P is given by the uniform distribution.
17 00:01:37,579 –> 00:01:44,533 Ok there you see, with this probability space we have the whole information of this random experiment.
18 00:01:44,733 –> 00:01:48,979 So all the possible outcomes with the probabilities are given here.
19 00:01:49,514 –> 00:01:56,267 However maybe we are in the situation where we are only interested in the sum of the two numbers the dice show.
20 00:01:56,771 –> 00:02:03,152 For example we could be in a game where this is important and the colours of the two dice don’t matter at all.
21 00:02:03,629 –> 00:02:08,082 Then exactly in such a case we would define a random variable.
22 00:02:08,629 –> 00:02:12,189 and as i told you before, we would simply call it X.
23 00:02:12,871 –> 00:02:20,771 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.
24 00:02:21,457 –> 00:02:26,176 So what X should do we already know. It should give us the sum of the 2 numbers.
25 00:02:26,643 –> 00:02:34,475 Hence a sample given by (omega_1, omega_2) is mapped to the sum omega_1 + omega_2.
26 00:02:34,675 –> 00:02:37,729 So you see, this is not a complicated map at all.
27 00:02:38,657 –> 00:02:44,535 What we can remember in this case is that the input is a sample and the output is a number.
28 00:02:44,735 –> 00:02:52,945 Ok, so this is a typical example of a random variable, where you see it’s simply extracts the information we are interested in.
29 00:02:53,514 –> 00:02:57,136 Therefore later we will work with a lot of random variables.
30 00:02:57,743 –> 00:03:01,800 However first i would say we have to give the correct definition now.
31 00:03:02,471 –> 00:03:07,250 and as promised, this is the general definition one uses in probability theory.
32 00:03:07,614 –> 00:03:13,187 Ok, so what we need are 2 spaces we call measurable spaces or event spaces.
33 00:03:13,771 –> 00:03:19,657 The first one should be given by a set. A sample space Omega with a corresponding sigma algebra A
34 00:03:20,271 –> 00:03:26,075 and then the second one is given by a set Omega tilde with a corresponding sigma algebra A tilde.
35 00:03:26,657 –> 00:03:32,610 So you could say we have here probability spaces, where the probability measure P is not fixed yet.
36 00:03:33,157 –> 00:03:40,485 Hence we are just interested in the events, the elements of a sigma algebra and therefore we talk of events spaces.
37 00:03:40,685 –> 00:03:46,102 However in measure theory we would call these spaces measurable spaces
38 00:03:46,302 –> 00:03:51,756 and in fact the random variable we define now, we would call a measurable map.
39 00:03:51,956 –> 00:03:57,681 Nevertheless as often in probability theory, we use some special names for these objects.
40 00:03:57,881 –> 00:04:05,334 Ok, now we consider a map we call capital X from one sample space Omega into the other one, Omega tilde
41 00:04:05,757 –> 00:04:12,009 and then this map is called a random variable if it is a measurable map in the measure theoretical sense.
42 00:04:12,486 –> 00:04:18,519 Which means we have to look at all the pre-images of the events in the second sample space, Omega tilde.
43 00:04:18,886 –> 00:04:24,141 Here lets denote an element of curved A tilde, just with a normal A tilde.
44 00:04:24,629 –> 00:04:29,491 and then we know, this pre-image of A tilde is the subset of Omega.
45 00:04:29,691 –> 00:04:35,971 However in the end, when we have a probability measure P we want to measure these sets here.
46 00:04:36,529 –> 00:04:43,536 Hence it’s necessary that this is not just a subset of Omega, but also an element of the sigma algebra A.
47 00:04:43,736 –> 00:04:49,425 Therefore this is exactly the right condition we need here for all A tilde.
48 00:04:49,625 –> 00:04:54,065 OK, there we have it. This is the whole definition of a random variable.
49 00:04:54,114 –> 00:04:56,141 A concept we will need a lot.
50 00:04:56,341 –> 00:05:00,011 Therefore i would say lets immediately look at some examples.
51 00:05:00,600 –> 00:05:05,057 So maybe for the start lets discuss the details of the random variable from above.
52 00:05:05,607 –> 00:05:09,329 There the first event space was given by (Omega, A).
53 00:05:09,529 –> 00:05:15,386 Where Omega is the sample space given by 1 to 6, squared and A is just the power set.
54 00:05:15,943 –> 00:05:20,244 Moreover the second event space was given by the real number line.
55 00:05:20,929 –> 00:05:23,486 Hence this would be Omega tilde.
56 00:05:24,192 –> 00:05:27,812 However now we should ask: What is A tilde?
57 00:05:28,257 –> 00:05:31,657 In the end i can already tell you it will not matter at all,
58 00:05:31,857 –> 00:05:36,543 but usually when we have the real number line, we would take the Borel sigma algebra.
59 00:05:37,057 –> 00:05:39,743 Therefore we also do this here
60 00:05:40,250 –> 00:05:47,103 and now i can ask you: is this map from before, this X, actually a random variable?
61 00:05:47,729 –> 00:05:53,886 At first glance it seems to be that we have to check a lot, because we need to check all the pre-images here.
62 00:05:54,386 –> 00:05:59,253 However please recall the sigma algebra A here is the whole power set.
63 00:05:59,886 –> 00:06:08,586 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.
64 00:06:09,025 –> 00:06:11,714 Which is of course trivially fulfilled.
65 00:06:12,167 –> 00:06:19,866 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.
66 00:06:20,371 –> 00:06:25,232 So in this case we can easily conclude that the map X is a random variable.
67 00:06:25,432 –> 00:06:30,740 Indeed most of the time we won’t have any problems at all fulfilling this condition here
68 00:06:30,940 –> 00:06:35,463 or to put it in other words counter examples are always very artificial.
69 00:06:35,829 –> 00:06:43,822 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.
70 00:06:44,022 –> 00:06:48,100 Hence instead of the largest one, the power set, we take the smallest one.
71 00:06:49,057 –> 00:06:54,294 So the only events we have in our sigma algebra A are the empty set and Omega itself.
72 00:06:55,229 –> 00:07:01,386 When of course we immediately find a counter example. You just have to look at the pre-image of the singleton 2.
73 00:07:01,986 –> 00:07:06,820 In words this means the sum of the 2 numbers of the dice is exactly 2.
74 00:07:07,200 –> 00:07:11,944 So there is only one dice throw possible. The 2 dice show both one.
75 00:07:12,271 –> 00:07:16,681 Therefore this pre-image is just a set with only one element.
76 00:07:17,257 –> 00:07:21,489 and that’s the reason i chose 2 here, because then i don’t have to write so much
77 00:07:21,689 –> 00:07:23,801 and of course you immediately see the result.
78 00:07:24,001 –> 00:07:28,498 This set is neither the empty set nor the whole set Omega.
79 00:07:29,157 –> 00:07:32,583 So it’s not an element in the sigma algebra A.
80 00:07:32,783 –> 00:07:37,272 and then we can conclude, in this case X is not a random variable.
81 00:07:37,986 –> 00:07:43,479 Ok, now in summary what you should see here is: random variables are not complicated at all.
82 00:07:43,814 –> 00:07:48,811 and indeed most of the time the fact that we have a random variable is immediately given.
83 00:07:49,011 –> 00:07:53,030 Ok now i want to close this video with an important notation.
84 00:07:53,614 –> 00:07:58,741 Assume again, that we have 2 measurable spaces also called event spaces.
85 00:07:59,257 –> 00:08:01,829 Moreover lets also fix two other things.
86 00:08:02,060 –> 00:08:05,329 First we have a random variable X as before
87 00:08:05,720 –> 00:08:09,519 and secondly maybe not so surprising, we have a probability measure P.
88 00:08:09,719 –> 00:08:13,036 Defined on the first event space on the left.
89 00:08:13,236 –> 00:08:18,461 This means when we take any set A tilde from the sigma algebra A tilde
90 00:08:18,661 –> 00:08:23,931 and look at the pre-image under X, then we can calculate the probability of this event.
91 00:08:24,486 –> 00:08:31,326 Simply because we know by the definition of a random variable that this set lies in the sigma algebra A.
92 00:08:32,071 –> 00:08:34,867 Hence P of this set makes sense.
93 00:08:35,300 –> 00:08:39,571 Therefore one usually uses a shorter, but strange notation for this.
94 00:08:40,486 –> 00:08:44,849 One simply writes P of X in A tilde.
95 00:08:45,429 –> 00:08:50,563 First it looks a little bit odd, but you will see this a lot in probability theory
96 00:08:51,071 –> 00:08:55,714 and indeed it makes a little bit more sense when we use the definition of a pre-image.
97 00:08:56,057 –> 00:09:00,344 Which is simply the set of all lower case omega in capital Omega
98 00:09:01,029 –> 00:09:05,491 with the property that X of lower case omega lies in A tilde.
99 00:09:06,043 –> 00:09:11,309 Hence you can see the left hand side as a shortcut for writing this whole set.
100 00:09:11,509 –> 00:09:19,076 The important thing you should remember here is that this literally does not make sense, but it stands for a whole set in Omega.
101 00:09:19,276 –> 00:09:24,193 Please note in the same sense also other shortcuts as this are used as well.
102 00:09:24,686 –> 00:09:28,149 In more detail we will discuss this in the next video.
103 00:09:28,349 –> 00:09:32,043 Therefore i hope i see there and have a nice day. Bye!
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Quiz Content
Q1: What is not a correct name for $(\Omega, \mathcal{A})$ where $\Omega$ is a set and $\mathcal{A}$ is a $\sigma$-algebra on $\Omega$.
A1: measure space
A2: event space
A3: measurable space
Q2: Let $(\Omega, \mathcal{A})$ and $(\widetilde{\Omega}, \widetilde{\mathcal{A}})$ be two measurable space. What is the correct definition of a random variable?
A1: $X : \Omega \rightarrow \Omega$ with $\omega \mapsto 1$
A2: $X : \Omega \rightarrow \widetilde{\Omega}$ with $X^{-1}(\widetilde{A}) \in \mathcal{A}$ for all $\widetilde{A} \in \widetilde{\mathcal{A}}$
A3: $X : \widetilde{\Omega} \rightarrow \widetilde{\Omega}$ with $X^{-1}(\widetilde{A}) \in \mathcal{A}$ for some $\widetilde{A} \in \widetilde{\mathcal{A}}$
A4: $X : \Omega \rightarrow \widetilde{\Omega}$ with $X(\widetilde{A}) \in \mathcal{A}$ for all $\widetilde{A} \in \widetilde{\mathcal{A}}$
Q3: Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X \colon \Omega \rightarrow \mathbb{R}$ be a random variable. What is the meaning of the notation $P(X \in B)$?
A1: $\mathbb{P}({ \omega \in \Omega \mid X(\omega) \in B })$
A2: $\mathbb{P}({ \omega \in \mathbb{R} \mid X(\omega) \subseteq B })$
A3: $\mathbb{P}(X(B))$
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Last update: 2024-10
