1
00:00:01,864 --> 00:00:04,030
PROFESSOR: We just saw some
random variables come up

2
00:00:04,030 --> 00:00:05,426
in the bigger number game.

3
00:00:05,426 --> 00:00:07,800
And we're going to be talking
now about random variables,

4
00:00:07,800 --> 00:00:10,180
just formally what they
are and their definition

5
00:00:10,180 --> 00:00:11,990
of independence for
random variables.

6
00:00:11,990 --> 00:00:15,100
But let's begin by looking
at the informal idea.

7
00:00:15,100 --> 00:00:17,010
Again, a random variable
is a number that's

8
00:00:17,010 --> 00:00:19,280
produced by a random process.

9
00:00:19,280 --> 00:00:21,650
So a typical example
that comes up

10
00:00:21,650 --> 00:00:23,480
where you get a random
variable is you've

11
00:00:23,480 --> 00:00:25,420
got some system
that you're watching

12
00:00:25,420 --> 00:00:29,510
and you're going to time it to
see when the next crash comes,

13
00:00:29,510 --> 00:00:30,750
if it crashes.

14
00:00:30,750 --> 00:00:34,000
So assuming that
this is unpredictable

15
00:00:34,000 --> 00:00:35,850
that it happens in
some random way,

16
00:00:35,850 --> 00:00:39,430
then the number of hours from
the present until the next time

17
00:00:39,430 --> 00:00:43,090
the system crashes
is a number that's

18
00:00:43,090 --> 00:00:45,150
produced by this
random process of

19
00:00:45,150 --> 00:00:48,000
whether the system works or not.

20
00:00:48,000 --> 00:00:50,000
Number of faulty
pixels in a monitor.

21
00:00:50,000 --> 00:00:52,190
When you're building
the monitors

22
00:00:52,190 --> 00:00:57,160
and delivering them to the
actual computer manufacturers,

23
00:00:57,160 --> 00:00:59,410
there's a certain
probability that some

24
00:00:59,410 --> 00:01:02,570
of the millions of pixels in the
monitor are going to be faulty.

25
00:01:02,570 --> 00:01:05,489
And you could think of
that number of pixels

26
00:01:05,489 --> 00:01:08,680
is also produced from an
unpredictable randomness

27
00:01:08,680 --> 00:01:12,370
in the manufacturing process.

28
00:01:12,370 --> 00:01:14,864
One that really is modeled
in physics as random

29
00:01:14,864 --> 00:01:16,280
is when you have
a Geiger counter,

30
00:01:16,280 --> 00:01:17,960
you're measuring
alpha particles.

31
00:01:17,960 --> 00:01:20,250
The number of alpha
particles that

32
00:01:20,250 --> 00:01:23,560
are detected by a given
Geiger counter in a second

33
00:01:23,560 --> 00:01:27,800
is believed to be
a random number.

34
00:01:27,800 --> 00:01:29,290
There's a distribution
that it has

35
00:01:29,290 --> 00:01:30,890
but the number of
alpha particles

36
00:01:30,890 --> 00:01:34,530
is not always the same
from second to second,

37
00:01:34,530 --> 00:01:37,180
and so it's a random variable.

38
00:01:37,180 --> 00:01:41,090
And finally, we'll look at
the standard abstract example

39
00:01:41,090 --> 00:01:42,330
of flipping coins.

40
00:01:42,330 --> 00:01:45,140
And if I flip coins
then the number

41
00:01:45,140 --> 00:01:47,130
of heads in a given
number of flips--

42
00:01:47,130 --> 00:01:49,550
let's say I flip a coin n times.

43
00:01:49,550 --> 00:01:52,300
The number of heads
will be another rather

44
00:01:52,300 --> 00:01:53,590
standard random variable.

45
00:01:53,590 --> 00:01:57,134
OK what is abstractly
a random variable?

46
00:01:57,134 --> 00:01:58,800
Oops, I'm getting
ahead of myself again.

47
00:01:58,800 --> 00:02:02,330
Let's look at that example
of three fair coins.

48
00:02:02,330 --> 00:02:06,740
So each coin has a probability
of being heads that's a half

49
00:02:06,740 --> 00:02:08,100
and tails being a half.

50
00:02:08,100 --> 00:02:09,600
I'm going to flip
the three of them.

51
00:02:09,600 --> 00:02:11,808
And I'm going to assume that
they're distinguishable.

52
00:02:11,808 --> 00:02:15,690
So there's a first coin, a
second coin, and a third coin.

53
00:02:15,690 --> 00:02:18,800
Or alternatively you could think
of flipping the same coin three

54
00:02:18,800 --> 00:02:20,380
times.

55
00:02:20,380 --> 00:02:25,362
So the number of
heads is a number

56
00:02:25,362 --> 00:02:27,820
that comes out of this random
process of flipping the three

57
00:02:27,820 --> 00:02:28,610
coins.

58
00:02:28,610 --> 00:02:31,320
So it's a number that's
either from 0 to 3.

59
00:02:31,320 --> 00:02:33,240
There could be no
heads or all heads.

60
00:02:33,240 --> 00:02:37,070
So it is a basic example
of a random variable

61
00:02:37,070 --> 00:02:39,040
where you're producing
this integer based

62
00:02:39,040 --> 00:02:40,680
on how the coins flip.

63
00:02:40,680 --> 00:02:44,210
Another one is simply a
[? 0-1 ?] valued random

64
00:02:44,210 --> 00:02:49,470
variable where it signals 1 if
all 3 coins match in what they

65
00:02:49,470 --> 00:02:56,260
come up with, and 0
if they don't match.

66
00:02:56,260 --> 00:02:58,830
Now once I have these
random variables defined,

67
00:02:58,830 --> 00:03:01,060
one of the things
that's a convenient use

68
00:03:01,060 --> 00:03:02,940
of random variables is
to use them to define

69
00:03:02,940 --> 00:03:04,530
various kinds of events.

70
00:03:04,530 --> 00:03:07,420
So the event that
C equals 1, that's

71
00:03:07,420 --> 00:03:10,900
an event that-- it's
a set of outcomes

72
00:03:10,900 --> 00:03:14,980
where the count is 1 and it
has a certain probability.

73
00:03:14,980 --> 00:03:17,330
This is the event
of exactly 1 head.

74
00:03:17,330 --> 00:03:20,990
There are 3 possible outcomes
among the 8 outcomes of heads

75
00:03:20,990 --> 00:03:22,640
and tails with 3 coins.

76
00:03:22,640 --> 00:03:26,760
So it has probability 3/8.

77
00:03:26,760 --> 00:03:28,510
I could also just
talk about the outcome

78
00:03:28,510 --> 00:03:30,790
that C is greater
than or equal to 1.

79
00:03:30,790 --> 00:03:33,870
Well C is greater
than or equal to 1

80
00:03:33,870 --> 00:03:37,001
when there is at least 1 head.

81
00:03:37,001 --> 00:03:39,500
Or put another way, the only
time that C is not greater than

82
00:03:39,500 --> 00:03:43,300
or equal to 1 is when
you have all tails.

83
00:03:43,300 --> 00:03:46,350
So there's a 7/8 chance,
7 out of 8 outcomes

84
00:03:46,350 --> 00:03:47,934
involve 1 or more heads.

85
00:03:47,934 --> 00:03:50,100
So the probability that C
greater than or equal to 1

86
00:03:50,100 --> 00:03:51,930
is 7/8.

87
00:03:51,930 --> 00:03:53,510
Here's a weirder one.

88
00:03:53,510 --> 00:03:56,700
I can use the two variables
C and M to define an event.

89
00:03:56,700 --> 00:04:00,440
What's the probability that
C times M is greater than 0?

90
00:04:00,440 --> 00:04:02,880
Well since C and M
are both non-negative

91
00:04:02,880 --> 00:04:06,760
variables, the probability
that their product is greater

92
00:04:06,760 --> 00:04:10,250
than 0 is equal to the
probability that each of them

93
00:04:10,250 --> 00:04:12,410
is greater than 0.

94
00:04:12,410 --> 00:04:15,490
OK, what does it mean
that M is greater

95
00:04:15,490 --> 00:04:17,089
than 0 and C is greater than 0?

96
00:04:17,089 --> 00:04:19,040
Well it says there's at
least 1 head-- that's

97
00:04:19,040 --> 00:04:20,680
what C greater than 0 means.

98
00:04:20,680 --> 00:04:24,200
And M greater than 0
means all the coins match.

99
00:04:24,200 --> 00:04:28,000
This is an obscure way of
describing the event all heads,

100
00:04:28,000 --> 00:04:31,850
and it has a course
probability 1/8.

101
00:04:31,850 --> 00:04:33,640
Now we come to the
formal definition.

102
00:04:33,640 --> 00:04:35,970
So formally, a random
variable is simply

103
00:04:35,970 --> 00:04:41,950
a function that maps outcomes
in the sample space to numbers.

104
00:04:41,950 --> 00:04:45,180
We think of the outcomes
in the sample space

105
00:04:45,180 --> 00:04:48,560
as the results of a
random experiment.

106
00:04:48,560 --> 00:04:51,080
They are an outcome and
they have a probability.

107
00:04:51,080 --> 00:04:55,410
And when the outcome is
translated into a real number

108
00:04:55,410 --> 00:04:57,000
that you think of
as being produced

109
00:04:57,000 --> 00:04:59,160
as a result of that
outcome, that's

110
00:04:59,160 --> 00:05:00,840
what the random variable does.

111
00:05:00,840 --> 00:05:03,760
So formally, a random
variable is not a variable.

112
00:05:03,760 --> 00:05:06,450
Or it's a function
that maps the sample

113
00:05:06,450 --> 00:05:08,099
space to the real numbers.

114
00:05:08,099 --> 00:05:09,640
And it's got to be
total, by the way.

115
00:05:09,640 --> 00:05:11,040
It's a total function.

116
00:05:11,040 --> 00:05:15,220
Usually this would be a
real valued random variable.

117
00:05:15,220 --> 00:05:16,710
Usually it's the real numbers.

118
00:05:16,710 --> 00:05:20,270
Might be a subset of the real
numbers like the integer valued

119
00:05:20,270 --> 00:05:21,250
random variables.

120
00:05:21,250 --> 00:05:26,100
Occasionally we'll use complex
valued random variables.

121
00:05:26,100 --> 00:05:29,390
Actually, that happens
in physics a good deal

122
00:05:29,390 --> 00:05:33,040
in quantum mechanics,
but not for our purposes.

123
00:05:33,040 --> 00:05:36,010
We're just going to mean
real value from now on when

124
00:05:36,010 --> 00:05:39,490
we talk about random variables.

125
00:05:39,490 --> 00:05:42,610
So abstractly or intuitively
what the random variable

126
00:05:42,610 --> 00:05:46,220
is doing really is it
just packaging together

127
00:05:46,220 --> 00:05:48,720
in one object R,
the random variable,

128
00:05:48,720 --> 00:05:51,710
a whole bunch of events that
are defined by the value

129
00:05:51,710 --> 00:05:52,710
that R takes.

130
00:05:52,710 --> 00:05:56,120
So for every possible real
number, if I look at the event

131
00:05:56,120 --> 00:05:59,370
that R is equal to a,
that's an interesting event.

132
00:05:59,370 --> 00:06:05,110
And it's one of the basic
events that R puts together.

133
00:06:05,110 --> 00:06:07,880
And if you knew the
answer to all of these R

134
00:06:07,880 --> 00:06:13,730
equals a's, then you
really know a lot about R.

135
00:06:13,730 --> 00:06:16,760
And with this
understanding that R

136
00:06:16,760 --> 00:06:19,920
is a package of
events of the form R

137
00:06:19,920 --> 00:06:23,290
is equal to a, then a lot
of the event properties

138
00:06:23,290 --> 00:06:25,350
carry right over to
random variables directly.

139
00:06:25,350 --> 00:06:28,590
That's why this little topic
of introducing random variables

140
00:06:28,590 --> 00:06:31,590
is also about independence
because the definition

141
00:06:31,590 --> 00:06:33,190
of independence
carries right over.

142
00:06:33,190 --> 00:06:35,870
Namely, a bunch of
random variables

143
00:06:35,870 --> 00:06:40,120
are mutually independent if
the events that they define

144
00:06:40,120 --> 00:06:42,880
are all mutually independent.

145
00:06:42,880 --> 00:06:46,180
So if and only if the
events that are-- each event

146
00:06:46,180 --> 00:06:50,540
defined by R1 and R2 and
through Rn, that set of events

147
00:06:50,540 --> 00:06:52,700
are mutually independent
no matter what

148
00:06:52,700 --> 00:06:57,480
the values are chosen that we
decide to look at for R1 and R2

149
00:06:57,480 --> 00:06:57,980
through Rn.

150
00:07:01,040 --> 00:07:02,760
And of course there's
an alternative way

151
00:07:02,760 --> 00:07:04,540
we could always express
independent events

152
00:07:04,540 --> 00:07:13,960
in terms of products instead
of conditional probabilities.

153
00:07:13,960 --> 00:07:16,370
So we could say-- or
instead of invoking

154
00:07:16,370 --> 00:07:18,060
the idea of mutual
independence we

155
00:07:18,060 --> 00:07:20,800
could say explicitly where
it comes from as an equation.

156
00:07:20,800 --> 00:07:25,270
It means that the probability
that R1 is equal to a1 and R2

157
00:07:25,270 --> 00:07:29,030
is equal to a1 and
Rn is equal to an

158
00:07:29,030 --> 00:07:31,640
is equal to the product
of the probabilities--

159
00:07:31,640 --> 00:07:33,040
of the individual
probabilities--

160
00:07:33,040 --> 00:07:36,760
that R1 is a1 times the
probability of R2 is a2.

161
00:07:36,760 --> 00:07:39,020
And the definition then
of mutual independence

162
00:07:39,020 --> 00:07:41,360
of the random
variables R1 through n,

163
00:07:41,360 --> 00:07:44,880
Rn holds is that
this equation it

164
00:07:44,880 --> 00:07:51,120
holds for all possible values,
little a1 through little an.

165
00:07:51,120 --> 00:07:52,220
So let's just practice.

166
00:07:52,220 --> 00:07:55,540
Are the variables C, which
is the count of the number

167
00:07:55,540 --> 00:07:58,120
of heads when you flip
three coins, and M,

168
00:07:58,120 --> 00:08:01,240
[? the 0-1 ?] valued random
variable that tells you whether

169
00:08:01,240 --> 00:08:04,150
there's a match, are
they independent?

170
00:08:04,150 --> 00:08:08,520
Well certainly not,
because there's definitely

171
00:08:08,520 --> 00:08:12,160
a positive probability that
the count will be 1 that you'll

172
00:08:12,160 --> 00:08:13,810
get at least a head.

173
00:08:13,810 --> 00:08:15,610
And there's a
positive probability

174
00:08:15,610 --> 00:08:16,794
that they all will match.

175
00:08:16,794 --> 00:08:18,210
It's the probability
of a quarter.

176
00:08:18,210 --> 00:08:21,229
So the product of
those 2 is positive,

177
00:08:21,229 --> 00:08:23,520
but of course the probability
that you match and you'll

178
00:08:23,520 --> 00:08:27,720
have exactly 1 head is 0 because
if you have exactly 1 head

179
00:08:27,720 --> 00:08:30,980
you must have 2 tails
and there's no match.

180
00:08:30,980 --> 00:08:34,520
So without thinking
very hard about what

181
00:08:34,520 --> 00:08:37,669
the probabilities are
we can immediately

182
00:08:37,669 --> 00:08:41,440
see that the product is not
equal to the probability

183
00:08:41,440 --> 00:08:44,680
of the conjunction or the
and, and therefore they're

184
00:08:44,680 --> 00:08:47,330
not independent.

185
00:08:47,330 --> 00:08:50,032
Well here's one that's a
little bit more interesting.

186
00:08:50,032 --> 00:08:51,740
In order to explain
it I've got to set up

187
00:08:51,740 --> 00:08:54,400
the idea of an indicator
variable, which itself

188
00:08:54,400 --> 00:08:55,590
is a very important concept.

189
00:08:55,590 --> 00:08:58,830
So if I have an event
A, I can package A

190
00:08:58,830 --> 00:09:00,410
into a random variable.

191
00:09:00,410 --> 00:09:03,240
Just like the match random
variable was really packaging

192
00:09:03,240 --> 00:09:06,660
the event that the coins
matched into a [? 0-1 ?] valued

193
00:09:06,660 --> 00:09:09,940
variable, I'm going to define
the indicator variable for any

194
00:09:09,940 --> 00:09:14,620
event A to be 1 if A occurs
and 0 if A does not occur.

195
00:09:14,620 --> 00:09:17,750
So now I'm able to
capture everything

196
00:09:17,750 --> 00:09:22,020
that matters about event A
by the random variable IA.

197
00:09:22,020 --> 00:09:24,540
If I have IA I know what
A is, and if I have A I

198
00:09:24,540 --> 00:09:26,260
know what IA is.

199
00:09:26,260 --> 00:09:28,800
And it means that
really I can think

200
00:09:28,800 --> 00:09:31,500
of events as special
cases of random variables.

201
00:09:31,500 --> 00:09:35,020
Now when you do this
you need a sanity check.

202
00:09:35,020 --> 00:09:37,640
Because remember we've
defined independence

203
00:09:37,640 --> 00:09:39,730
of random variables one way.

204
00:09:39,730 --> 00:09:42,070
I mean it's a concept
of independence that

205
00:09:42,070 --> 00:09:43,630
holds for random variables.

206
00:09:43,630 --> 00:09:45,480
We have another
concept of independence

207
00:09:45,480 --> 00:09:46,500
that holds for events.

208
00:09:46,500 --> 00:09:48,710
Now the definition
for random variable

209
00:09:48,710 --> 00:09:51,110
was motivated by the
definition for events

210
00:09:51,110 --> 00:09:53,890
but it's a different
definition of independence

211
00:09:53,890 --> 00:09:55,500
of different kinds of objects.

212
00:09:55,500 --> 00:09:58,100
Now if this correspondence
between events and indicator

213
00:09:58,100 --> 00:10:01,590
variables is going to make
sense and not confuse us

214
00:10:01,590 --> 00:10:07,010
it should be the case that
two events are independent if

215
00:10:07,010 --> 00:10:10,570
and only if their indicator
variables are independent.

216
00:10:10,570 --> 00:10:13,960
That is, IA and IB are
independent if and only

217
00:10:13,960 --> 00:10:16,430
if the events A and
B are independent.

218
00:10:16,430 --> 00:10:18,220
And this is a lovely
little exercise.

219
00:10:18,220 --> 00:10:20,880
It's like a three-line
proof for you to verify.

220
00:10:20,880 --> 00:10:23,440
I'm not going bother
to do it on the slide

221
00:10:23,440 --> 00:10:24,940
because it's good practice.

222
00:10:24,940 --> 00:10:27,860
So this would be a
moment to stop and verify

223
00:10:27,860 --> 00:10:30,040
that using the two
definitions of independence,

224
00:10:30,040 --> 00:10:32,760
the definition of what
it means for IA and IB

225
00:10:32,760 --> 00:10:34,970
to be independent
as random variables

226
00:10:34,970 --> 00:10:36,880
and comparing that to
the definition of what

227
00:10:36,880 --> 00:10:40,080
it means for A and B to
be independent as events,

228
00:10:40,080 --> 00:10:40,755
they match.

229
00:10:44,720 --> 00:10:49,560
If we look at the event
of an odd number of heads

230
00:10:49,560 --> 00:10:54,690
we can ask now
whether the event M,

231
00:10:54,690 --> 00:10:56,970
which is the indicator
variable for a match--

232
00:10:56,970 --> 00:11:00,610
the random variable M-- and
the indicator variable IO

233
00:11:00,610 --> 00:11:01,960
are dependent or not.

234
00:11:01,960 --> 00:11:05,570
Now both of these depend
on all the three coins.

235
00:11:05,570 --> 00:11:08,060
IO is looking at
all 3 coins to see

236
00:11:08,060 --> 00:11:09,560
if there are an odd
number of heads,

237
00:11:09,560 --> 00:11:12,890
M is looking at all 3
coins to see if they're

238
00:11:12,890 --> 00:11:14,220
all heads or all tails.

239
00:11:14,220 --> 00:11:18,970
And it's not clear with all
that common basis for returning

240
00:11:18,970 --> 00:11:20,260
what value they have.

241
00:11:20,260 --> 00:11:23,530
It's not immediately obvious
that they're independent,

242
00:11:23,530 --> 00:11:25,340
but as a matter
of fact they are.

243
00:11:25,340 --> 00:11:27,447
And again this is
absolutely something

244
00:11:27,447 --> 00:11:28,530
that you should check out.

245
00:11:28,530 --> 00:11:30,720
If you don't stop the
video now to work it out,

246
00:11:30,720 --> 00:11:33,070
you should definitely
do it afterward.

247
00:11:33,070 --> 00:11:34,530
It's an important
little exercise

248
00:11:34,530 --> 00:11:35,488
and it's easy to check.

249
00:11:35,488 --> 00:11:40,460
All you have to do is check
that the probabilities

250
00:11:40,460 --> 00:11:45,250
of the event of odd number of
heads in the event all match

251
00:11:45,250 --> 00:11:46,680
are independent as events.

252
00:11:46,680 --> 00:11:50,700
Or you could use the
random variable definition

253
00:11:50,700 --> 00:11:53,890
and check that these two
random variables were

254
00:11:53,890 --> 00:11:56,600
independent by
checking 4 equations

255
00:11:56,600 --> 00:11:58,250
because this can
have values 0 and 1.

256
00:11:58,250 --> 00:12:02,720
And this can have value 0 and 1.

257
00:12:02,720 --> 00:12:08,050
Remember with independent
events we had the idea

258
00:12:08,050 --> 00:12:10,100
that if A was independent
of B it really

259
00:12:10,100 --> 00:12:12,350
meant that A was independent
of everything about B.

260
00:12:12,350 --> 00:12:15,800
In particular it was independent
of the complement of B as well.

261
00:12:15,800 --> 00:12:18,620
And a similar property
holds for random variables.

262
00:12:18,620 --> 00:12:21,240
So intuitively if R
is independent of S

263
00:12:21,240 --> 00:12:23,840
then R is really independent
of any information

264
00:12:23,840 --> 00:12:26,960
at all that you have
about S. And that

265
00:12:26,960 --> 00:12:28,840
can be made more
precise that R is

266
00:12:28,840 --> 00:12:31,210
independent of any
information about S

267
00:12:31,210 --> 00:12:35,270
by saying pick an arbitrary
function that maps R to R,

268
00:12:35,270 --> 00:12:36,410
total function.

269
00:12:36,410 --> 00:12:40,690
So what I can do is
think of f as giving me

270
00:12:40,690 --> 00:12:43,600
some information
about the value of S.

271
00:12:43,600 --> 00:12:46,930
So if R is independent
of S then in fact R

272
00:12:46,930 --> 00:12:51,510
is independent of f of S,
any transformation of S

273
00:12:51,510 --> 00:12:53,360
by a fixed non-random function.

274
00:12:56,400 --> 00:12:58,930
And of course the notion
of k-way independence

275
00:12:58,930 --> 00:13:01,810
carries right over
from the event case.

276
00:13:01,810 --> 00:13:06,270
If I have k random-- if I have
a bunch of random variables,

277
00:13:06,270 --> 00:13:08,940
a large number much
more than k, they're

278
00:13:08,940 --> 00:13:12,340
k-way independent if
every set of k of them

279
00:13:12,340 --> 00:13:14,210
are mutually independent.

280
00:13:14,210 --> 00:13:17,160
And of course as with
events we use the 2-way case

281
00:13:17,160 --> 00:13:20,200
to call them
pairwise independent.

282
00:13:20,200 --> 00:13:23,660
Again, we saw an example of
this in terms of events already

283
00:13:23,660 --> 00:13:26,600
but we can rephrase it now in
terms of indicator variables.

284
00:13:26,600 --> 00:13:30,730
If we let Hi be the
indicator variable for a head

285
00:13:30,730 --> 00:13:36,310
on a flip i-- of the
i flip of a coin--

286
00:13:36,310 --> 00:13:42,110
where i ranges from 1
through k, if we have k coins

287
00:13:42,110 --> 00:13:45,150
and Hi is the indicator
variable for how coin

288
00:13:45,150 --> 00:13:49,880
I came out, whether
or not there's a head,

289
00:13:49,880 --> 00:13:52,190
now O can be nicely expressed.

290
00:13:52,190 --> 00:13:54,540
The notion that there's an
odd number of heads is simply

291
00:13:54,540 --> 00:13:56,800
the mod 2 sum of the Hi's.

292
00:13:56,800 --> 00:14:00,380
And this by the way, is a trick
that we'll be using regularly

293
00:14:00,380 --> 00:14:03,800
that events now can be
defined rather nicely

294
00:14:03,800 --> 00:14:06,850
in terms of doing
operations on the arithmetic

295
00:14:06,850 --> 00:14:08,630
values of indicator variables.

296
00:14:08,630 --> 00:14:10,610
So O is nothing
but the mod 2 sum

297
00:14:10,610 --> 00:14:15,790
of the values of the indicator
variables Hi from 1 to k.

298
00:14:15,790 --> 00:14:20,710
And what we saw when we were
working with their event

299
00:14:20,710 --> 00:14:25,035
version is that any k of
these events are independent.

300
00:14:25,035 --> 00:14:26,180
I've got k plus 1.

301
00:14:26,180 --> 00:14:28,610
There's k Hi's and
there's O, which

302
00:14:28,610 --> 00:14:31,320
makes the k plus
1-- k plus first.

303
00:14:31,320 --> 00:14:44,047
[AUDIO OUT] And the reason why
any k of them were independent

304
00:14:44,047 --> 00:14:46,380
was discussed in the previous
slide when we were looking

305
00:14:46,380 --> 00:14:49,260
at the events of there
being an odd number of heads

306
00:14:49,260 --> 00:14:52,920
and a head coming
up on the i flip.

307
00:14:55,850 --> 00:14:58,490
The reason why pairwise
independence gets singled out

308
00:14:58,490 --> 00:15:01,880
is that we'll see that for a
bunch of major applications

309
00:15:01,880 --> 00:15:05,910
this pairwise
independence is sufficient

310
00:15:05,910 --> 00:15:08,702
and rather than verifying
mutual independence.

311
00:15:08,702 --> 00:15:10,410
It's harder to check
mutual independence.

312
00:15:10,410 --> 00:15:14,530
You've got a lot more
equations to check.

313
00:15:14,530 --> 00:15:18,130
And in fact it often doesn't
hold in circumstances where

314
00:15:18,130 --> 00:15:19,660
pairwise does hold.

315
00:15:19,660 --> 00:15:20,860
So this is good to know.

316
00:15:20,860 --> 00:15:23,520
We'll be making use of
it in an application

317
00:15:23,520 --> 00:15:29,640
later when we look at sampling
and the law of large numbers.