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Title: Probability Measures
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Series: Probability Theory
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YouTube-Title: Probability Theory 2 | Probability Measures
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Bright video: https://youtu.be/u5IouBwYji4
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Dark video: https://youtu.be/Z_jJXumponM
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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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Exercise Download PDF sheets
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Subtitle on GitHub: pt02_sub_eng.srt
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Timestamps
00:00 Idea of a probability measure
01:40 Requirements
04:19 Sigma algebra
05:50 Sigma additivity
06:31 Definition probability measure
07:22 Example
08:57 Exercise about properties of probability measures
09:30 Outro
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Subtitle in English
1 00:00:00,660 –> 00:00:03,760 Hello and welcome back to probability theory
2 00:00:04,300 –> 00:00:08,880 And first i want to thank all the nice people that support this channel on Steady or Paypal.
3 00:00:09,430 –> 00:00:13,510 Now in todays part 2 we will talk about probability measures.
4 00:00:13,980 –> 00:00:18,280 Of course this will be a notion that will be important through the whole series
5 00:00:18,480 –> 00:00:23,380 because we can put the whole randomness we want to describe into such a probability measure.
6 00:00:24,150 –> 00:00:29,840 Now if you already know some measure theory you also already know what probability measures are.
7 00:00:30,620 –> 00:00:35,300 Indeed in short you can just say these are the measure with total mass 1.
8 00:00:36,160 –> 00:00:39,840 However of course i want to explain a little bit more about this here.
9 00:00:40,600 –> 00:00:45,300 Maybe a good starting point is always to think of a rectangle or a square in the plane.
10 00:00:46,210 –> 00:00:50,310 This is the overall set we consider and usually we call it omega
11 00:00:51,300 –> 00:00:56,580 and also you should know in probability theory this is usually known as the sample space.
12 00:00:57,430 –> 00:01:04,849 Therefore this omega could be any set, because we just put all possible outcomes of our random experiment into it.
13 00:01:05,550 –> 00:01:11,330 and then we just visualize this set with the rectangle here where the whole area should be 1.
14 00:01:11,940 –> 00:01:15,830 Simply because the probability to get any outcome should be 1.
15 00:01:16,310 –> 00:01:22,510 And now a probability measure should also tell you the probability for a given subset.
16 00:01:23,880 –> 00:01:27,370 So what we want here is simply a map we call “P”.
17 00:01:27,470 –> 00:01:29,970 That maps subsets to numbers.
18 00:01:30,770 –> 00:01:34,550 Hence this curved “A” here should be a collection of subsets.
19 00:01:35,290 –> 00:01:41,780 In other words the best case scenario would be that we have all the subsets of omega in this fancy “A”.
20 00:01:42,570 –> 00:01:49,160 Ok, now for getting a useful probability notion here we really have some important requirements for this map.
21 00:01:49,980 –> 00:01:55,370 The first thing we already said. We want That the probability of the whole sample space should be 1.
22 00:01:56,290 –> 00:02:01,140 So when we measure probabilities the maximum value we want to get out should be 1.
23 00:02:01,850 –> 00:02:05,140 On the other hand the minimal value should be 0.
24 00:02:05,830 –> 00:02:12,210 Therefore no matter which subset “A” we put in we always want to get out a number between 0 and 1.
25 00:02:12,790 –> 00:02:15,770 Hence we have the closed 0,1 interval here.
26 00:02:16,670 –> 00:02:22,360 Going back to our rectangle here this means the smallest area we could measure should be 0.
27 00:02:23,050 –> 00:02:27,850 However all other values between 0 and 1 could be possible for areas.
28 00:02:28,410 –> 00:02:32,300 Ok then in the next step let’s look at another subset “B” here.
29 00:02:33,320 –> 00:02:38,610 Now if there’s no overlap between both sets “A” and “B” we could simply add the areas.
30 00:02:39,170 –> 00:02:47,550 This means that the probability of the subset “A union B” should be just the sum of both probabilities.
31 00:02:48,230 –> 00:02:53,450 So we want this nice and natural formula in the case that “A” and “B” are disjoint.
32 00:02:54,100 –> 00:02:58,890 As a reminder this means that the intersection of both subsets should be the empty set.
33 00:02:59,430 –> 00:03:03,390 Speaking of the empty set. What should be the area of the empty set?
34 00:03:04,080 –> 00:03:07,980 Of course the only useful definition should be to choose it as zero.
35 00:03:08,580 –> 00:03:12,980 So the probability to get no outcome at all should always be 0.
36 00:03:13,630 –> 00:03:18,620 Ok, now we have almost everything we need here. There is only one thing missing.
37 00:03:19,200 –> 00:03:23,190 Namely this formula should also hold in a limit process.
38 00:03:23,730 –> 00:03:27,360 This means that we look here at a countable union of subsets.
39 00:03:27,980 –> 00:03:34,360 In our picture this would mean that we approximate an area by adding up countable many areas.
40 00:03:34,850 –> 00:03:39,340 Hence in our formula we will have a series from one to infinity.
41 00:03:40,310 –> 00:03:42,410 So this union goes to a sum.
42 00:03:43,270 –> 00:03:48,070 Of course for this formula we also need the assumption that we have a disjoint union here.
43 00:03:48,730 –> 00:03:53,120 In other words you would say the family “A_j” is pairwise disjoint.
44 00:03:53,730 –> 00:04:00,530 This means that no matter which two different sets “A_i” and “A_j” you choose you always have an empty intersection.
45 00:04:01,230 –> 00:04:07,600 So with this we have the requirements we demand when we want to define a general notion of a probability.
46 00:04:08,540 –> 00:04:14,200 Now to satisfy these claims here we need for the domain “A” a so called sigma-algebra.
47 00:04:14,820 –> 00:04:18,660 Therefore lets talk about the definition of a sigma-algebra.
48 00:04:19,279 –> 00:04:26,720 I don’t want to go into the details, because i have a whole series about measure theory, where the first videos are about sigma-algebras.
49 00:04:27,360 –> 00:04:30,580 The overall idea is the same. For any set omega
50 00:04:30,950 –> 00:04:33,530 we only want to consider suitable subsets.
51 00:04:34,170 –> 00:04:37,409 However these subsets together should fulfill some rules.
52 00:04:38,380 –> 00:04:42,120 So we take a collection of subsets denoted with this curved “A”
53 00:04:42,320 –> 00:04:46,010 Which means we have a subset of the power set of omega.
54 00:04:46,470 –> 00:04:53,460 Now thinking of a probability problem in this curved “A” we have all the events we want the probability for.
55 00:04:53,900 –> 00:04:59,380 For example for the throw of a die we have the event of getting an even number as an outcome
56 00:04:59,990 –> 00:05:03,930 and this event is a subset of which we want to measure the probability.
57 00:05:04,750 –> 00:05:08,850 Therefore in this case we need that event to be an element of this “A” here.
58 00:05:09,570 –> 00:05:16,570 However what we always want to do is to measure the probability of the empty set and the whole sample space omega.
59 00:05:17,220 –> 00:05:20,120 Hence both sets should be an element of “A”.
60 00:05:21,240 –> 00:05:27,000 Then the next property we need is, if we take any subset “A” that lies in our curved “A”
61 00:05:27,340 –> 00:05:30,050 then the complement should also be in curved “A”.
62 00:05:30,630 –> 00:05:35,620 The complement in general is defined as omega, the whole space, without the set “A”
63 00:05:35,810 –> 00:05:38,630 and i always denote it with this upper index “c”.
64 00:05:39,330 –> 00:05:43,930 Please note this makes sense, because if we have the probability for one event “A”
65 00:05:44,100 –> 00:05:48,240 then we should also be able to calculate the probability of not “A”.
66 00:05:48,770 –> 00:05:55,650 Ok and then the third property is not so hard to understand, because you already know we want this for the probability measure.
67 00:05:56,390 –> 00:06:03,830 So we just take countable many sets from “A” and then the whole union should also be an element of “A”.
68 00:06:04,960 –> 00:06:09,820 Now having these three properties the collection “A” is called a sigma algebra
69 00:06:10,490 –> 00:06:14,450 and usually we will write it in this way with a lowercase sigma.
70 00:06:15,060 –> 00:06:22,630 Also important to note here is that the elements of a sigma algebra in probability theory are usually called events.
71 00:06:23,520 –> 00:06:31,030 Ok now such a sigma algebra is as i told you before the domain for such a probability measure we now define.
72 00:06:31,730 –> 00:06:36,300 Therefore lets fix a sample space omega and a sigma algebra “A”.
73 00:06:36,940 –> 00:06:43,630 Then a map we now call “P” with domain “A” and codomain the interval [0,1]
74 00:06:44,350 –> 00:06:48,140 is called a probability measure if it fulfills two properties
75 00:06:48,770 –> 00:06:52,810 and because we have already discussed them above we can just copy them now.
76 00:06:53,340 –> 00:07:00,310 The first one is simply that the probability of the whole space is 1 and the probability of the empty event is 0.
77 00:07:00,900 –> 00:07:04,470 And the second one is what we call sigma additivity.
78 00:07:05,060 –> 00:07:09,890 And please remember for this sigma additivity we need pairwise disjoint sets.
79 00:07:10,480 –> 00:07:17,720 Ok and with this you know the general notion of a probability measure and of course we will use that a lot here.
80 00:07:18,320 –> 00:07:21,460 However maybe lets start with a simple example.
81 00:07:22,210 –> 00:07:25,410 I would say lets look a the die from last video again.
82 00:07:25,980 –> 00:07:30,480 So if we just throw it one time we immediately know the sample space omega.
83 00:07:31,260 –> 00:07:35,200 All possible outcomes from 1 to 6 form our sample space.
84 00:07:35,690 –> 00:07:39,680 However now for the sigma algebra “A” you already have the choice.
85 00:07:40,430 –> 00:07:43,930 The questions here is in how many events could you be interested.
86 00:07:44,420 –> 00:07:49,520 The simple answer would be we are interested in all events. Therefore we choose the power set.
87 00:07:50,130 –> 00:07:57,030 Often i would say this is a useful approach. However it won’t work in the case that the sample space is infinite,
88 00:07:57,390 –> 00:08:01,800 but also there one could simply say i choose a large enough sigma algebra
89 00:08:02,220 –> 00:08:06,420 and then we define the probability measure on this sigma algebra.
90 00:08:07,030 –> 00:08:12,910 In our case, because we have an ordinary die we have that each side has the same probability.
91 00:08:13,630 –> 00:08:21,300 Therefore in our general definition here this would mean that we count the elements of “A” and devide by the elements of omega.
92 00:08:21,960 –> 00:08:26,460 Of course in this case this would mean in the denominator we have the number 6.
93 00:08:27,020 –> 00:08:34,010 For example if we want to calculate the probability of the event of throwing a 2 we get out 1 over 6.
94 00:08:34,669 –> 00:08:40,770 On the other hand the probability of the event of throwing an even number is 3 divided by 6
95 00:08:41,299 –> 00:08:42,970 or in other words one half.
96 00:08:43,429 –> 00:08:49,520 Ok and now here you see this is our mathematical model of throwing one ordinary die
97 00:08:50,190 –> 00:08:56,800 and in later videos you will see we can easily expand this whole model to cover also other examples.
98 00:08:57,240 –> 00:09:02,620 Ok now for closing this video i give you a small exercise about probability measures.
99 00:09:03,030 –> 00:09:08,110 Try to prove that for a general probability measure “P” and an event “A”
100 00:09:08,600 –> 00:09:13,340 the probability of the complement is 1 minus the probability of “A”.
101 00:09:13,990 –> 00:09:19,280 So this is a very nice property that immediately comes out of the two other properties here.
102 00:09:20,000 –> 00:09:26,300 Ok i think that’s good enough for today in the next videos we will cover a lot of other examples of probability measures.
103 00:09:26,880 –> 00:09:30,520 Therefore i hope i see you there and have a nice day. Bye!
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Quiz Content
Q1: A $\sigma$-algebra is a subset of the power set $\mathcal{P}(\Omega)$.
A1: True.
A2: False.
Q2: Which of these $\mathcal{A}$ is a $\sigma$-algebra for $\Omega = \mathbb{N}$?
A1: $\mathcal{A} = \Omega$
A2: $\mathcal{A} = { { \emptyset }, { \Omega } }$
A3: $\mathcal{A} = { \emptyset, \Omega }$
A4: $\mathcal{A} = { \emptyset }$
Q3: Is $\mathbb{P}: \mathcal{P}(\mathbb{R}) \rightarrow \mathbb{R}$ with $\mathbb{P}(A) = 0$ for all sets $A$ a probability measure?
A1: Yes, it is.
A2: No, one property is not satisfied.
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Last update: 2024-10
