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Title: Distribution of a Random Variable
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Series: Probability Theory
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YouTube-Title: Probability Theory 11 | Distribution of a Random Variable
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Subtitle in English
1 00:00:00,460 –> 00:00:02,279 Hello and welcome back to
2 00:00:02,289 –> 00:00:04,070 probability theory
3 00:00:04,619 –> 00:00:06,030 and as always, first, I want
4 00:00:06,039 –> 00:00:07,380 to thank all the nice people
5 00:00:07,389 –> 00:00:08,329 that support this channel
6 00:00:08,340 –> 00:00:09,619 on Steady or PayPal.
7 00:00:10,140 –> 00:00:12,000 And today in part 11, we
8 00:00:12,010 –> 00:00:13,479 will talk about the distribution
9 00:00:13,489 –> 00:00:14,829 of a random variable.
10 00:00:15,529 –> 00:00:16,940 So therefore, please first
11 00:00:16,950 –> 00:00:18,479 recall, a random variable
12 00:00:18,489 –> 00:00:20,479 acts between two event spaces.
13 00:00:20,569 –> 00:00:22,309 It’s given as a map capital
14 00:00:22,319 –> 00:00:23,940 X from Omega to
15 00:00:23,950 –> 00:00:24,840 Omega tilde
16 00:00:25,540 –> 00:00:27,270 and in fact, the most important
17 00:00:27,280 –> 00:00:28,600 random variables occur,
18 00:00:28,610 –> 00:00:30,069 when Omega tilde is given
19 00:00:30,079 –> 00:00:31,989 as the real number line R
20 00:00:33,400 –> 00:00:34,919 and then a useful choice
21 00:00:34,930 –> 00:00:36,479 for the Sigma algebra is
22 00:00:36,490 –> 00:00:37,860 given by the Borel Sigma
23 00:00:37,869 –> 00:00:38,540 algebra.
24 00:00:39,319 –> 00:00:39,740 OK.
25 00:00:39,750 –> 00:00:41,060 Now, these are the random
26 00:00:41,069 –> 00:00:42,340 variables we will talk about
27 00:00:42,349 –> 00:00:42,919 today.
28 00:00:43,709 –> 00:00:45,159 The overall idea for today
29 00:00:45,169 –> 00:00:46,650 is that on the one hand,
30 00:00:46,659 –> 00:00:48,049 you have to think of an abstract
31 00:00:48,060 –> 00:00:49,680 probability space given by
32 00:00:49,689 –> 00:00:50,279 Omega
33 00:00:50,930 –> 00:00:52,290 and on the other hand, we
34 00:00:52,299 –> 00:00:53,750 have a very concrete probability
35 00:00:53,759 –> 00:00:55,650 space given by the real number
36 00:00:55,659 –> 00:00:56,090 line.
37 00:00:56,759 –> 00:00:58,430 I said probability spaces,
38 00:00:58,439 –> 00:00:59,909 because this is what we will
39 00:00:59,919 –> 00:01:00,840 have in the end.
40 00:01:00,849 –> 00:01:02,310 But of course, first, we
41 00:01:02,319 –> 00:01:03,830 will start just with event
42 00:01:03,840 –> 00:01:04,550 spaces.
43 00:01:05,110 –> 00:01:07,040 So most importantly, a probability
44 00:01:07,050 –> 00:01:08,099 measure on the right-hand
45 00:01:08,110 –> 00:01:09,620 side is still missing.
46 00:01:10,309 –> 00:01:11,500 Now corresponding to the
47 00:01:11,510 –> 00:01:12,809 two event spaces
48 00:01:12,819 –> 00:01:14,239 I would say we have a random
49 00:01:14,269 –> 00:01:15,540 variable we call capital
50 00:01:15,550 –> 00:01:16,169 X.
51 00:01:16,319 –> 00:01:17,809 From the last video, you
52 00:01:17,819 –> 00:01:19,489 already know this map has
53 00:01:19,500 –> 00:01:21,120 one property we call
54 00:01:21,129 –> 00:01:22,010 measurable
55 00:01:22,720 –> 00:01:24,110 and soon you will see why
56 00:01:24,120 –> 00:01:25,410 we really need that.
57 00:01:26,139 –> 00:01:27,720 However, first, I would say
58 00:01:27,730 –> 00:01:29,639 let’s add a probability measure
59 00:01:29,650 –> 00:01:30,989 on the left-hand side.
60 00:01:31,580 –> 00:01:33,260 So we have an abstract probability
61 00:01:33,269 –> 00:01:35,099 measure P defined on the
62 00:01:35,110 –> 00:01:36,430 Sigma algebra A.
63 00:01:37,010 –> 00:01:38,449 However, now we are not
64 00:01:38,459 –> 00:01:39,680 interested in abstract
65 00:01:39,690 –> 00:01:41,019 calculations here on the
66 00:01:41,029 –> 00:01:42,910 left-hand side, because our
67 00:01:42,919 –> 00:01:44,870 whole problem is given with
68 00:01:44,879 –> 00:01:46,319 this random variable X.
69 00:01:47,000 –> 00:01:48,139 So we would rather like to
70 00:01:48,150 –> 00:01:49,389 calculate here on the right-
71 00:01:49,400 –> 00:01:50,750 hand side with real
72 00:01:50,760 –> 00:01:51,419 numbers.
73 00:01:52,209 –> 00:01:53,629 Hence, what we want to add
74 00:01:53,639 –> 00:01:55,190 is a probability measure
75 00:01:55,199 –> 00:01:56,550 for the real number line
76 00:01:56,559 –> 00:01:58,099 for the Borel Sigma algebra.
77 00:01:58,669 –> 00:02:00,610 So we introduce a new probability
78 00:02:00,620 –> 00:02:02,470 measure and since it corresponds
79 00:02:02,480 –> 00:02:04,139 to the random viable X, we
80 00:02:04,150 –> 00:02:05,510 call it P_X.
81 00:02:06,389 –> 00:02:07,930 Of course, this is a nice
82 00:02:07,940 –> 00:02:09,320 picture you really should
83 00:02:09,330 –> 00:02:11,210 always have in mind, but
84 00:02:11,220 –> 00:02:12,830 it does not tell us what
85 00:02:12,839 –> 00:02:14,289 the definition of P_X
86 00:02:14,300 –> 00:02:14,839 is.
87 00:02:15,619 –> 00:02:17,190 Therefore, let’s do this
88 00:02:17,199 –> 00:02:18,160 in a definition.
89 00:02:18,929 –> 00:02:19,320 Here
90 00:02:19,330 –> 00:02:21,160 we will explain what we mean
91 00:02:21,169 –> 00:02:22,960 when we say distribution
92 00:02:22,970 –> 00:02:24,509 of a random variable X
93 00:02:25,190 –> 00:02:26,750 and in fact, this is a very
94 00:02:26,759 –> 00:02:27,770 general definition.
95 00:02:27,779 –> 00:02:29,589 It works for any probability
96 00:02:29,600 –> 00:02:31,130 space given by a set
97 00:02:31,139 –> 00:02:33,089 Omega, a Sigma algebra A,
98 00:02:33,270 –> 00:02:34,570 and the probability measure
99 00:02:34,580 –> 00:02:35,009 P
100 00:02:35,759 –> 00:02:36,990 and the only other thing
101 00:02:37,000 –> 00:02:38,539 we need is a random variable
102 00:02:38,550 –> 00:02:40,520 X from Omega into
103 00:02:40,529 –> 00:02:42,339 R. Of course, as
104 00:02:42,350 –> 00:02:43,990 before the Sigma algebra,
105 00:02:44,000 –> 00:02:45,419 we choose for the real number
106 00:02:45,429 –> 00:02:46,990 line is the Borel Sigma
107 00:02:47,000 –> 00:02:47,610 algebra.
108 00:02:48,440 –> 00:02:50,360 So you see the whole assumptions
109 00:02:50,369 –> 00:02:51,979 fit with this picture here.
110 00:02:53,000 –> 00:02:54,369 Therefore, in the next step,
111 00:02:54,380 –> 00:02:56,080 we can define the map
112 00:02:56,089 –> 00:02:56,949 P_X.
113 00:02:57,850 –> 00:02:59,089 Which should have the Borel
114 00:02:59,100 –> 00:03:00,740 Sigma algebra as its
115 00:03:00,750 –> 00:03:01,339 domain.
116 00:03:02,070 –> 00:03:03,259 And because it should be
117 00:03:03,270 –> 00:03:04,130 a probability measure
118 00:03:04,139 –> 00:03:05,729 in the end, we can say it
119 00:03:05,740 –> 00:03:06,529 maps into
120 00:03:06,539 –> 00:03:07,899 [0,1].
121 00:03:08,100 –> 00:03:09,660 Now to define the map, let’s
122 00:03:09,669 –> 00:03:11,610 take any Borel set B
123 00:03:11,839 –> 00:03:13,509 and write P_X of
124 00:03:13,520 –> 00:03:15,449 B is equal to
125 00:03:16,380 –> 00:03:17,990 P, the abstract
126 00:03:18,000 –> 00:03:19,339 P, of
127 00:03:20,089 –> 00:03:21,509 the preimage of
128 00:03:21,520 –> 00:03:23,149 B under X.
129 00:03:24,050 –> 00:03:25,539 So you see this is a very
130 00:03:25,550 –> 00:03:27,130 natural construction when
131 00:03:27,139 –> 00:03:28,910 we have any set here and
132 00:03:28,919 –> 00:03:30,589 want to measure it, we pull
133 00:03:30,600 –> 00:03:32,300 it back to the original space
134 00:03:32,309 –> 00:03:33,869 here and measure it with
135 00:03:33,880 –> 00:03:34,270 P.
136 00:03:35,110 –> 00:03:36,970 And then this whole construction
137 00:03:36,979 –> 00:03:38,809 defines a new measure here
138 00:03:38,820 –> 00:03:39,929 on the right-hand side.
139 00:03:40,710 –> 00:03:42,389 Now, as a reminder, you already
140 00:03:42,399 –> 00:03:44,149 know there is another notation
141 00:03:44,160 –> 00:03:45,740 for denoting the preimage
142 00:03:45,750 –> 00:03:47,100 in probability theory.
143 00:03:47,750 –> 00:03:49,619 One simply writes X
144 00:03:49,630 –> 00:03:50,369 in B.
145 00:03:51,360 –> 00:03:52,710 Hence, you could also see
146 00:03:52,720 –> 00:03:54,660 this as a definition for
147 00:03:54,669 –> 00:03:55,500 P_X.
148 00:03:56,169 –> 00:03:56,669 OK.
149 00:03:56,679 –> 00:03:58,429 Now this map P_X
150 00:03:58,460 –> 00:04:00,309 is what we call the distribution
151 00:04:00,320 –> 00:04:02,059 of the random variable X.
152 00:04:02,550 –> 00:04:03,690 More concretely one would
153 00:04:03,699 –> 00:04:05,660 say probability distribution
154 00:04:05,669 –> 00:04:07,460 of X and
155 00:04:07,470 –> 00:04:08,759 sometimes you also see the
156 00:04:08,770 –> 00:04:10,169 long name probability
157 00:04:10,179 –> 00:04:11,919 distribution measure of X.
158 00:04:12,779 –> 00:04:14,000 It all means the same.
159 00:04:14,009 –> 00:04:15,350 Namely this map
160 00:04:15,440 –> 00:04:16,279 P_X.
161 00:04:17,070 –> 00:04:18,869 However, of course, the important
162 00:04:18,880 –> 00:04:20,690 part here is, this actually
163 00:04:20,700 –> 00:04:22,369 defines a probability measure
164 00:04:22,380 –> 00:04:23,149 on R.
165 00:04:23,679 –> 00:04:25,320 So let’s put that into a
166 00:04:25,329 –> 00:04:27,250 proposition. So
167 00:04:27,260 –> 00:04:28,700 you see this is a good point
168 00:04:28,709 –> 00:04:30,290 to refresh your memory,
169 00:04:30,299 –> 00:04:31,970 What we need for a probability
170 00:04:31,980 –> 00:04:33,970 measure. Essentially, we
171 00:04:33,980 –> 00:04:35,450 just have two properties,
172 00:04:35,459 –> 00:04:36,980 where the actual hard one
173 00:04:36,989 –> 00:04:38,309 is the so-called sigma
174 00:04:38,320 –> 00:04:39,149 additivity.
175 00:04:39,809 –> 00:04:41,220 Now, of course, in the proof
176 00:04:41,230 –> 00:04:42,829 we can use, that we already
177 00:04:42,839 –> 00:04:44,730 know that this P, the blue
178 00:04:44,739 –> 00:04:46,649 one, is already a probability
179 00:04:46,660 –> 00:04:47,109 measure.
180 00:04:47,899 –> 00:04:49,420 And the other part we need
181 00:04:49,429 –> 00:04:51,410 to use is that X is a random
182 00:04:51,420 –> 00:04:51,929 variable.
183 00:04:51,940 –> 00:04:53,250 So it’s measurable.
184 00:04:53,970 –> 00:04:54,359 OK.
185 00:04:54,369 –> 00:04:56,140 Then I would say let’s start
186 00:04:56,149 –> 00:04:57,510 with a simple fact
187 00:04:58,029 –> 00:04:59,750 Namely that the preimage
188 00:04:59,760 –> 00:05:01,589 of the whole set on the right,
189 00:05:01,600 –> 00:05:03,320 is the whole set on the left-
190 00:05:03,329 –> 00:05:03,950 hand side.
191 00:05:04,500 –> 00:05:06,100 So this is simply Omega
192 00:05:06,109 –> 00:05:07,450 and of course, this holds
193 00:05:07,459 –> 00:05:09,190 for every map from Omega
194 00:05:09,200 –> 00:05:10,019 into R.
195 00:05:10,750 –> 00:05:12,040 And with this fact, we can
196 00:05:12,049 –> 00:05:13,630 calculate P_X of
197 00:05:13,640 –> 00:05:15,570 R. By definition
198 00:05:15,579 –> 00:05:17,369 it’s simply P of the pre-
199 00:05:17,489 –> 00:05:19,010 image of R, which is
200 00:05:19,019 –> 00:05:19,570 Omega.
201 00:05:20,040 –> 00:05:21,890 So we have P of Omega
202 00:05:21,899 –> 00:05:23,799 which is one, because P
203 00:05:23,809 –> 00:05:25,250 is a probability measure.
204 00:05:26,019 –> 00:05:27,739 So you see this was not
205 00:05:27,750 –> 00:05:29,000 complicated at all
206 00:05:29,529 –> 00:05:31,359 and now we can do exactly
207 00:05:31,369 –> 00:05:33,149 the same for the empty set.
208 00:05:34,059 –> 00:05:35,239 Not so surprising.
209 00:05:35,250 –> 00:05:36,750 The preimage of the empty
210 00:05:36,760 –> 00:05:38,420 set is always the empty
211 00:05:38,429 –> 00:05:38,839 set.
212 00:05:39,459 –> 00:05:39,760 OK.
213 00:05:39,769 –> 00:05:41,190 Then I would say let’s shorten
214 00:05:41,200 –> 00:05:42,859 the whole calculation, because
215 00:05:42,869 –> 00:05:44,519 it works the same as above.
216 00:05:45,000 –> 00:05:46,549 In the end, we get P of the
217 00:05:46,559 –> 00:05:48,390 empty set, which is zero
218 00:05:48,399 –> 00:05:50,070 by the definition of a probability
219 00:05:50,079 –> 00:05:50,519 measure.
220 00:05:51,200 –> 00:05:51,700 OK.
221 00:05:51,709 –> 00:05:53,100 And with this, we have already
222 00:05:53,109 –> 00:05:54,559 proven half of the things
223 00:05:54,570 –> 00:05:56,359 we need for having a probability
224 00:05:56,369 –> 00:05:56,769 measure.
225 00:05:57,399 –> 00:05:59,140 Hence, only the Sigma
226 00:05:59,149 –> 00:06:01,079 additivity remains to show.
227 00:06:01,779 –> 00:06:03,299 In order to prove this, we
228 00:06:03,309 –> 00:06:04,660 need to choose countable
229 00:06:04,670 –> 00:06:05,869 many Borel sets.
230 00:06:05,880 –> 00:06:07,470 So B_1 B_2 and so
231 00:06:07,480 –> 00:06:09,410 on, and you know they
232 00:06:09,420 –> 00:06:11,220 should be pairwise disjoint.
233 00:06:11,380 –> 00:06:12,700 So the intersection with
234 00:06:12,709 –> 00:06:14,579 any two sets is always
235 00:06:14,589 –> 00:06:15,529 the empty set.
236 00:06:16,480 –> 00:06:17,880 Now, the good thing we have
237 00:06:17,890 –> 00:06:19,420 is that this stays also
238 00:06:19,429 –> 00:06:21,410 true for the preimages.
239 00:06:21,940 –> 00:06:23,440 Again, this is a fact that
240 00:06:23,450 –> 00:06:25,089 holds for any map from
241 00:06:25,100 –> 00:06:26,299 Omega into R,
242 00:06:26,970 –> 00:06:28,790 because the preimage has
243 00:06:28,799 –> 00:06:30,410 the general property that
244 00:06:30,420 –> 00:06:31,609 it is stable under
245 00:06:31,619 –> 00:06:32,649 intersections.
246 00:06:33,529 –> 00:06:35,100 In other words, we can pull
247 00:06:35,109 –> 00:06:36,619 in the intersection here.
248 00:06:37,440 –> 00:06:39,390 And now by assumption B_i
249 00:06:39,579 –> 00:06:41,459 and B_j are disjoint,
250 00:06:41,649 –> 00:06:43,329 this is the empty set
251 00:06:43,910 –> 00:06:45,059 and then we have the same
252 00:06:45,070 –> 00:06:46,899 as before. The preimage
253 00:06:46,910 –> 00:06:48,540 of the empty set is the
254 00:06:48,549 –> 00:06:49,489 empty set.
255 00:06:50,010 –> 00:06:51,519 Hence the conclusion here
256 00:06:51,529 –> 00:06:53,359 is, these countable many
257 00:06:53,369 –> 00:06:55,290 preimages are also
258 00:06:55,299 –> 00:06:56,649 pairwise disjoint.
259 00:06:57,339 –> 00:06:58,829 Of course, this is a very
260 00:06:58,839 –> 00:07:00,679 nice fact, but we should not
261 00:07:00,690 –> 00:07:02,380 forget that by the definition
262 00:07:02,390 –> 00:07:04,329 of a random variable, these
263 00:07:04,339 –> 00:07:06,100 preimages lie in the
264 00:07:06,109 –> 00:07:07,429 sigma algebra A
265 00:07:08,290 –> 00:07:09,910 and this is needed when we
266 00:07:09,920 –> 00:07:11,880 want to use the Sigma additivity
267 00:07:12,040 –> 00:07:13,540 for the original probability
268 00:07:13,549 –> 00:07:14,390 measure P
269 00:07:15,079 –> 00:07:16,970 and of course, this is exactly
270 00:07:16,980 –> 00:07:18,420 what we want to do now.
271 00:07:19,200 –> 00:07:20,779 So what we should write down
272 00:07:20,790 –> 00:07:22,459 is P_X of the
273 00:07:22,470 –> 00:07:23,940 infinite union of
274 00:07:23,950 –> 00:07:24,799 B_J.
275 00:07:25,829 –> 00:07:27,329 Then in the next step as
276 00:07:27,339 –> 00:07:29,170 before we use the definition
277 00:07:29,179 –> 00:07:30,309 of P_X.
278 00:07:30,690 –> 00:07:32,609 So now we have P
279 00:07:32,619 –> 00:07:34,429 of the preimage of the infinite
280 00:07:34,440 –> 00:07:34,929 union.
281 00:07:35,739 –> 00:07:37,160 Hence, what we want to use
282 00:07:37,170 –> 00:07:39,000 here is again a general
283 00:07:39,010 –> 00:07:40,559 fact for preimages.
284 00:07:41,339 –> 00:07:42,690 In particular, I can tell
285 00:07:42,700 –> 00:07:44,619 you the preimage is stable
286 00:07:44,630 –> 00:07:46,369 under any unions.
287 00:07:47,149 –> 00:07:48,579 In other words, we are allowed
288 00:07:48,589 –> 00:07:50,329 to pull out the union here
289 00:07:51,100 –> 00:07:52,540 and there you see we have
290 00:07:52,549 –> 00:07:54,440 what we want. A union of
291 00:07:54,450 –> 00:07:56,059 pairwise disjoint sets
292 00:07:56,130 –> 00:07:57,140 inside P.
293 00:07:57,690 –> 00:07:59,540 Hence, finally, we use the
294 00:07:59,549 –> 00:08:01,459 Sigma additivity for P.
295 00:08:01,940 –> 00:08:03,480 So the whole thing is the
296 00:08:03,489 –> 00:08:05,440 infinite sum of P
297 00:08:05,450 –> 00:08:07,369 of the preimage of B_J
298 00:08:07,380 –> 00:08:08,359 under X.
299 00:08:09,079 –> 00:08:10,779 Which we can translate back
300 00:08:10,790 –> 00:08:11,820 to P_X.
301 00:08:12,489 –> 00:08:14,109 This results in the infinite
302 00:08:14,119 –> 00:08:15,619 sum of P_X
303 00:08:15,630 –> 00:08:16,220 (B_J)
304 00:08:16,859 –> 00:08:18,329 and there you see this is
305 00:08:18,339 –> 00:08:19,890 exactly what we wanted to
306 00:08:19,899 –> 00:08:20,480 prove.
307 00:08:21,279 –> 00:08:23,100 We have the Sigma additivity for
308 00:08:23,109 –> 00:08:25,070 P_X and also the two
309 00:08:25,079 –> 00:08:26,760 other rules here, such that
310 00:08:26,769 –> 00:08:28,519 we get a probability measure.
311 00:08:29,190 –> 00:08:30,279 So please keep that fact
312 00:08:30,290 –> 00:08:31,859 in mind. The distribution
313 00:08:31,869 –> 00:08:33,330 of a random variable is
314 00:08:33,340 –> 00:08:35,190 always a probability measure.
315 00:08:35,770 –> 00:08:37,429 And this is exactly what
316 00:08:37,440 –> 00:08:39,049 we wanted at the beginning.
317 00:08:39,729 –> 00:08:40,869 Now, before we talk about
318 00:08:40,879 –> 00:08:42,729 examples, I first have to
319 00:08:42,739 –> 00:08:44,087 tell you about an important
320 00:08:44,097 –> 00:08:44,749 notation,
321 00:08:45,619 –> 00:08:47,510 Namely, if we have a probability
322 00:08:47,520 –> 00:08:49,229 measure which we could call
323 00:08:49,239 –> 00:08:51,200 P tilde on the real number
324 00:08:51,210 –> 00:08:51,640 line
325 00:08:52,320 –> 00:08:53,820 and now we find out that
326 00:08:53,830 –> 00:08:55,289 the probability distribution
327 00:08:55,299 –> 00:08:57,020 of X, P_X, is
328 00:08:57,030 –> 00:08:58,869 exactly equal to this P
329 00:08:59,159 –> 00:08:59,450 tilde,
330 00:09:00,059 –> 00:09:01,719 then in this case, we
331 00:09:01,729 –> 00:09:02,950 write X tilde ~
332 00:09:03,380 –> 00:09:04,400 P tilde.
333 00:09:05,119 –> 00:09:06,979 Moreover, we read it as
334 00:09:06,989 –> 00:09:08,719 X is distributed as
335 00:09:08,729 –> 00:09:10,530 P tilde. Here
336 00:09:10,539 –> 00:09:12,099 please keep in mind, P tilde
337 00:09:12,109 –> 00:09:13,530 could be any probability
338 00:09:13,539 –> 00:09:15,400 measure. I call it P
339 00:09:15,580 –> 00:09:17,500 tilde simply to avoid any
340 00:09:17,510 –> 00:09:19,219 danger of confusion with
341 00:09:19,229 –> 00:09:21,140 the blue P, which is defined
342 00:09:21,150 –> 00:09:21,940 for Omega.
343 00:09:22,500 –> 00:09:24,479 However, here P_X and P
344 00:09:24,619 –> 00:09:26,299 tilde are probability measures
345 00:09:26,309 –> 00:09:27,659 for the real number line.
346 00:09:28,229 –> 00:09:28,700 OK.
347 00:09:28,710 –> 00:09:29,789 Now the last part of the
348 00:09:29,799 –> 00:09:31,270 video will be a nice
349 00:09:31,280 –> 00:09:32,099 example.
350 00:09:32,729 –> 00:09:33,849 Let’s take a very a common
351 00:09:33,859 –> 00:09:34,320 one.
352 00:09:34,330 –> 00:09:35,859 The flip of a coin.
353 00:09:36,559 –> 00:09:37,880 Here, the probability for
354 00:09:37,890 –> 00:09:39,559 getting heads should be lower
355 00:09:39,570 –> 00:09:40,510 case p
356 00:09:41,289 –> 00:09:43,039 and now our probability space
357 00:09:43,049 –> 00:09:44,770 should represent n tosses
358 00:09:44,780 –> 00:09:46,280 of the same coin here
359 00:09:46,309 –> 00:09:48,070 with order. OK?
360 00:09:48,080 –> 00:09:49,580 So you see this probability
361 00:09:49,590 –> 00:09:50,929 space is not complicated
362 00:09:50,940 –> 00:09:51,510 at all.
363 00:09:51,570 –> 00:09:53,440 The set Omega is given as
364 00:09:53,450 –> 00:09:55,219 the set {0.1} to the
365 00:09:55,229 –> 00:09:56,969 power n, where
366 00:09:56,979 –> 00:09:58,700 one should represent heads
367 00:09:58,750 –> 00:10:00,169 and zero tails
368 00:10:00,719 –> 00:10:02,080 and then the Sigma algebra
369 00:10:02,090 –> 00:10:03,679 is just the power set of
370 00:10:03,690 –> 00:10:04,250 Omega.
371 00:10:04,989 –> 00:10:06,289 Hence, the only question
372 00:10:06,299 –> 00:10:08,190 remains, what is the probability
373 00:10:08,200 –> 00:10:09,280 measure P here?
374 00:10:10,059 –> 00:10:11,739 Indeed, it has a name, it’s
375 00:10:11,750 –> 00:10:13,349 called the Bernoulli distribution
376 00:10:13,359 –> 00:10:15,030 with parameters n and
377 00:10:15,039 –> 00:10:15,489 p.
378 00:10:15,940 –> 00:10:17,599 Indeed the probability mass
379 00:10:17,609 –> 00:10:18,789 function you can immediately
380 00:10:18,799 –> 00:10:19,580 write down.
381 00:10:19,590 –> 00:10:21,270 It’s not complicated at all.
382 00:10:21,840 –> 00:10:23,630 We have factors p and
383 00:10:23,640 –> 00:10:25,489 1 minus p where p
384 00:10:25,500 –> 00:10:27,460 comes in for heads and 1
385 00:10:27,469 –> 00:10:28,830 minus p for tails.
386 00:10:29,559 –> 00:10:31,359 Hence, the power of p is
387 00:10:31,369 –> 00:10:33,140 the number of ones in Omega
388 00:10:33,299 –> 00:10:34,580 and the power of 1 minus
389 00:10:34,590 –> 00:10:36,570 p is the number of zeros
390 00:10:36,580 –> 00:10:37,260 in Omega.
391 00:10:38,020 –> 00:10:38,479 OK.
392 00:10:38,489 –> 00:10:40,210 Now, this probability space
393 00:10:40,219 –> 00:10:41,919 here, you should see as our
394 00:10:41,929 –> 00:10:43,510 abstract one on the left-
395 00:10:43,520 –> 00:10:45,469 hand side and the right-
396 00:10:45,479 –> 00:10:46,979 hand side, we only get when
397 00:10:46,989 –> 00:10:48,169 we define a random variable
398 00:10:48,679 –> 00:10:50,669 X and this
399 00:10:50,679 –> 00:10:52,030 one should be defined as
400 00:10:52,039 –> 00:10:53,780 counting the number of heads.
401 00:10:53,789 –> 00:10:55,119 So without order.
402 00:10:55,859 –> 00:10:57,219 In other words, when we use
403 00:10:57,229 –> 00:10:58,950 numbers, X of Omega
404 00:10:58,960 –> 00:11:00,260 should be defined as the
405 00:11:00,270 –> 00:11:01,719 number of ones in
406 00:11:01,729 –> 00:11:02,340 Omega.
407 00:11:03,070 –> 00:11:04,369 Now, if you want, you can
408 00:11:04,380 –> 00:11:06,309 rewatch part four where
409 00:11:06,320 –> 00:11:07,880 we already explained this
410 00:11:07,890 –> 00:11:09,210 random experiment here.
411 00:11:10,030 –> 00:11:11,489 Then it might not surprise
412 00:11:11,500 –> 00:11:12,849 you that the distribution
413 00:11:12,859 –> 00:11:14,510 of X is given by the
414 00:11:14,520 –> 00:11:15,909 binomial distribution.
415 00:11:16,549 –> 00:11:18,369 Indeed, if you do the explicit
416 00:11:18,380 –> 00:11:20,200 calculation for P_X,
417 00:11:20,330 –> 00:11:21,469 you see this
418 00:11:21,479 –> 00:11:22,690 coincides with the
419 00:11:22,700 –> 00:11:24,640 explanations we gave in part
420 00:11:24,650 –> 00:11:25,049 four.
421 00:11:26,000 –> 00:11:27,580 Hence, here you see this
422 00:11:27,590 –> 00:11:29,260 is a good example, where the
423 00:11:29,270 –> 00:11:30,919 actual random experiment
424 00:11:30,929 –> 00:11:32,500 we are interested in is
425 00:11:32,510 –> 00:11:34,169 hidden in a random variable
426 00:11:34,900 –> 00:11:36,500 and you will see, this will
427 00:11:36,510 –> 00:11:38,039 happen a lot in future.
428 00:11:38,799 –> 00:11:39,219 OK.
429 00:11:39,229 –> 00:11:40,440 Then in the next videos,
430 00:11:40,450 –> 00:11:42,200 we will continue our journey
431 00:11:42,210 –> 00:11:43,780 in probability theory.
432 00:11:44,299 –> 00:11:45,359 Therefore, I hope I see you
433 00:11:45,369 –> 00:11:46,739 there and have a nice day.
434 00:11:46,849 –> 00:11:47,650 Bye.
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Quiz Content
Q1: Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X \colon \Omega \rightarrow \mathbb{R}$ be a random variable. What is the definition of $\mathbb{P}_X$?
A1: $\mathbb{P}_X(B) = \mathbb{P}(X \leq B)$
A2: $\mathbb{P}_X(B) = \mathbb{P}(X(B))$
A3: $\mathbb{P}_X(B) = \mathbb{P}(X \geq B)$
A4: $\mathbb{P}_X(B) = \mathbb{P}(X \in B)$
A5: $\mathbb{P}_X(B) = \mathbb{P}(X \notin B)$
Q2: Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X \colon \Omega \rightarrow \mathbb{R}$ be a random variable. How do we call $\mathbb{P}_X$?
A1: Probability density function of $X$.
A2: Probability distribution of $X$.
A3: Cumulative distribution function of $X$.
A4: Measure space of $X$.
Q3: Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X \colon \Omega \rightarrow \mathbb{R}$ be a random variable. What is not correct for $\mathbb{P}_X$?
A1: It is a probability measure.
A2: It is a probability measure defined on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$.
A3: $\mathbb{P}_X(B) = \mathbb{P}(X^{-1}(B))$
A4: $\mathbb{P}_X(\Omega) = 1$
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Last update: 2024-10
