1
00:00:00,440 --> 00:00:03,540
Our discussion of random
variable so far has involved

2
00:00:03,540 --> 00:00:06,700
nothing but standard probability
calculations.

3
00:00:06,700 --> 00:00:09,780
Other than using the PMF
notation, we have

4
00:00:09,780 --> 00:00:11,240
done nothing new.

5
00:00:11,240 --> 00:00:14,670
It is now time to introduce a
truly new concept that plays a

6
00:00:14,670 --> 00:00:17,250
central role in probability
theory.

7
00:00:17,250 --> 00:00:20,750
This is the concept of the
expected value or expectation

8
00:00:20,750 --> 00:00:23,220
or mean of a random variable.

9
00:00:23,220 --> 00:00:26,930
It is a single number that
provides some kind of summary

10
00:00:26,930 --> 00:00:29,590
of a random variable
by telling us what

11
00:00:29,590 --> 00:00:31,430
it is on the average.

12
00:00:31,430 --> 00:00:34,000
Let us motivate with
an example.

13
00:00:34,000 --> 00:00:36,820
You play a game of chance
over and over, let

14
00:00:36,820 --> 00:00:39,120
us say 1,000 times.

15
00:00:39,120 --> 00:00:43,910
Each time that you play, you win
an amount of money, which

16
00:00:43,910 --> 00:00:47,350
is a random variable, and that
random variable takes the

17
00:00:47,350 --> 00:00:50,640
value 1, with probability 2/10,
the value of 2, with

18
00:00:50,640 --> 00:00:55,140
probability 5/10, the value of
4, with probability 3/10.

19
00:00:55,140 --> 00:00:58,920
You can plot the PMF of
this random variable.

20
00:00:58,920 --> 00:01:03,720
It takes values 1, 2, and 4.

21
00:01:03,720 --> 00:01:06,810
And the associated
probabilities are

22
00:01:06,810 --> 00:01:16,050
2/10, 5/10, and 3/10.

23
00:01:22,330 --> 00:01:25,860
How much do you expect to have
at the end of the day?

24
00:01:25,860 --> 00:01:29,170
Well, if you interpret
probabilities as frequencies,

25
00:01:29,170 --> 00:01:36,820
in a thousand plays, you expect
to have about 200 times

26
00:01:36,820 --> 00:01:42,650
this outcome to occur and this
outcome about 500 times and

27
00:01:42,650 --> 00:01:46,160
this outcome about 300 times.

28
00:01:46,160 --> 00:01:52,670
So your average gain is expected
to be your total

29
00:01:52,670 --> 00:01:59,800
gain, which is 1, 200
times, plus 2, 500

30
00:01:59,800 --> 00:02:05,110
times, plus 4, 300 times.

31
00:02:05,110 --> 00:02:06,430
This is your total gain.

32
00:02:06,430 --> 00:02:10,720
And to get to the average gain,
you divide by 1,000.

33
00:02:10,720 --> 00:02:14,400
And the expression that you get
can also be written in a

34
00:02:14,400 --> 00:02:25,480
simpler form as 1 times
2/10 plus 2 times 5/10

35
00:02:25,480 --> 00:02:29,880
plus 4 times 3/10.

36
00:02:29,880 --> 00:02:34,010
So this is what you expect to
get, on the average, if you

37
00:02:34,010 --> 00:02:35,340
keep playing that game.

38
00:02:38,470 --> 00:02:40,170
What have we done?

39
00:02:40,170 --> 00:02:44,640
We have calculated a certain
quantity which is a sort of

40
00:02:44,640 --> 00:02:48,690
average of the random variable
of interest.

41
00:02:48,690 --> 00:02:53,720
And what we did in this
summation here, we took each

42
00:02:53,720 --> 00:02:57,320
one of the possible values
of the random variable.

43
00:02:57,320 --> 00:02:59,470
Each possible value
corresponds to

44
00:02:59,470 --> 00:03:01,610
one term in the summation.

45
00:03:01,610 --> 00:03:05,260
And what we're adding is the
numerical value of the random

46
00:03:05,260 --> 00:03:08,970
variable times the probability
that this

47
00:03:08,970 --> 00:03:13,060
particular value is obtained.

48
00:03:13,060 --> 00:03:18,040
So when x is equal to 1, we
get 1 here and then the

49
00:03:18,040 --> 00:03:20,140
probability of 1.

50
00:03:20,140 --> 00:03:23,890
When we add the term
corresponding to x equals 2,

51
00:03:23,890 --> 00:03:28,320
we get little x equals to 2 and
next to it the probability

52
00:03:28,320 --> 00:03:32,090
that x is equal to
2, and so on.

53
00:03:32,090 --> 00:03:36,010
So this is what we call the
expected value of the random

54
00:03:36,010 --> 00:03:38,010
variable x.

55
00:03:38,010 --> 00:03:40,820
This is the formula that defines
it, but it's also

56
00:03:40,820 --> 00:03:44,170
important to always keep in
mind the interpretation of

57
00:03:44,170 --> 00:03:45,250
that formula.

58
00:03:45,250 --> 00:03:48,880
The expected value of a random
variable is to be interpreted

59
00:03:48,880 --> 00:03:53,980
as the average that you expect
to see in a large number of

60
00:03:53,980 --> 00:03:58,460
independent repetitions
of the experiment.

61
00:03:58,460 --> 00:04:03,100
One small technical caveat, if
we're dealing with a random

62
00:04:03,100 --> 00:04:07,780
variable that takes values in
a discrete but infinite set,

63
00:04:07,780 --> 00:04:11,110
this sum here is going
to be an infinite sum

64
00:04:11,110 --> 00:04:12,990
or an infinite series.

65
00:04:12,990 --> 00:04:16,320
And there's always a question
whether an infinite series has

66
00:04:16,320 --> 00:04:19,380
a well-defined limit or not.

67
00:04:19,380 --> 00:04:24,640
In order for it to have a
well-defined limit, we will be

68
00:04:24,640 --> 00:04:30,450
making the assumption that this
infinite series is, as

69
00:04:30,450 --> 00:04:34,480
it's called, absolutely
convergent, namely that if we

70
00:04:34,480 --> 00:04:37,159
replace the x's by their
absolute values--

71
00:04:37,159 --> 00:04:40,510
so we're adding positive
numbers,

72
00:04:40,510 --> 00:04:42,020
or nonnegative numbers--

73
00:04:42,020 --> 00:04:45,150
the sum of those numbers
is going to be finite.

74
00:04:45,150 --> 00:04:48,330
So this is a technical condition
that we need in

75
00:04:48,330 --> 00:04:52,060
order to make sure that this
expected value is a

76
00:04:52,060 --> 00:04:54,200
well-defined and finite
quantity.

77
00:04:57,930 --> 00:05:00,790
Let us now calculate the
expected value of a very

78
00:05:00,790 --> 00:05:04,160
simple random variable, the
Bernoulli random variable that

79
00:05:04,160 --> 00:05:07,770
takes the value 1 with
probability p and the value 0

80
00:05:07,770 --> 00:05:10,300
with probability 1 minus p.

81
00:05:10,300 --> 00:05:15,320
The expected value consists
of two terms.

82
00:05:15,320 --> 00:05:17,180
X can take the value 1.

83
00:05:17,180 --> 00:05:19,040
This happens with
probability p.

84
00:05:19,040 --> 00:05:21,540
Or it can take the
value of zero.

85
00:05:21,540 --> 00:05:24,300
This happens with probability
1 minus p.

86
00:05:24,300 --> 00:05:29,450
And therefore, the expected
value is just equal to p.

87
00:05:29,450 --> 00:05:34,409
As a special case, we may
consider the situation where X

88
00:05:34,409 --> 00:05:38,630
is the indicator random variable
of a certain event,

89
00:05:38,630 --> 00:05:47,450
A, so that X is equal to 1 if
and only if event A occurs.

90
00:05:47,450 --> 00:05:51,530
In this case, the probability
that X equals to 1, which is

91
00:05:51,530 --> 00:05:54,670
our parameter p, is the same
as the probability

92
00:05:54,670 --> 00:05:56,480
that event A occurs.

93
00:05:56,480 --> 00:06:00,170
And we have this relation.

94
00:06:00,170 --> 00:06:02,920
And so with this correspondence,
we readily

95
00:06:02,920 --> 00:06:06,840
conclude that the expected value
of an indicator random

96
00:06:06,840 --> 00:06:12,070
variable is equal to the
probability of that event.

97
00:06:12,070 --> 00:06:15,120
Let us move now to the
calculation of the expected

98
00:06:15,120 --> 00:06:18,720
value of a uniform
random variable.

99
00:06:18,720 --> 00:06:21,000
Let us consider, to keep
things simple, a random

100
00:06:21,000 --> 00:06:26,750
variable which is uniform
on the set from 0 to n.

101
00:06:26,750 --> 00:06:30,020
It's uniform, so the probability
of the values that

102
00:06:30,020 --> 00:06:33,590
it can take are all equal
to each other.

103
00:06:33,590 --> 00:06:37,550
It can take one of n plus 1
possible values, and so the

104
00:06:37,550 --> 00:06:41,860
probability of each one of the
values is 1 over n plus 1.

105
00:06:41,860 --> 00:06:43,909
We want to calculate
the expected value

106
00:06:43,909 --> 00:06:45,330
of this random variable.

107
00:06:45,330 --> 00:06:46,870
How do we proceed?

108
00:06:46,870 --> 00:06:49,730
We just recall the definition
of the expectation.

109
00:06:49,730 --> 00:06:54,090
It's a sum where we add over
all of the possible values.

110
00:06:54,090 --> 00:06:57,159
And for each one of the values,
we multiply by its

111
00:06:57,159 --> 00:06:58,409
corresponding probability.

112
00:07:04,270 --> 00:07:06,935
So we obtain a summation
of this form.

113
00:07:09,970 --> 00:07:15,370
We can factor out a factor of
1 over n plus 1, and we're

114
00:07:15,370 --> 00:07:20,930
left with 0 plus 1 plus
all the way up to n.

115
00:07:20,930 --> 00:07:25,840
And perhaps you remember the
formula for us summing those

116
00:07:25,840 --> 00:07:34,020
numbers, and it is n times
n plus 1 over 2.

117
00:07:34,020 --> 00:07:37,590
And after doing the
cancellations, we obtain a

118
00:07:37,590 --> 00:07:41,060
final answer, which
is n over 2.

119
00:07:41,060 --> 00:07:46,790
Incidentally, notice that n over
2 is just the midpoint of

120
00:07:46,790 --> 00:07:50,909
this picture that we have
here in this diagram.

121
00:07:50,909 --> 00:07:52,850
This is always the case.

122
00:07:52,850 --> 00:07:58,030
Whenever we have a PMF which is
symmetric around a certain

123
00:07:58,030 --> 00:08:00,940
point, then the expected
value will be

124
00:08:00,940 --> 00:08:03,060
the center of symmetry.

125
00:08:03,060 --> 00:08:07,060
More general, if you do not have
symmetry, the expected

126
00:08:07,060 --> 00:08:11,900
value turns out to be the center
of gravity of the PMF.

127
00:08:11,900 --> 00:08:15,600
If you think of these bars as
having weight, where the

128
00:08:15,600 --> 00:08:18,770
weight is proportional to their
height, the center of

129
00:08:18,770 --> 00:08:22,750
gravity is the point at which
you should put your finger if

130
00:08:22,750 --> 00:08:25,490
you want to balance that diagram
so that it doesn't

131
00:08:25,490 --> 00:08:30,060
fall in one direction
or the other.

132
00:08:30,060 --> 00:08:33,289
And we now close this segment
by providing one more

133
00:08:33,289 --> 00:08:36,039
interpretation of
expectations.

134
00:08:36,039 --> 00:08:39,130
Suppose that we have a class
consisting of n students and

135
00:08:39,130 --> 00:08:41,740
that the ith student
has a weight which

136
00:08:41,740 --> 00:08:44,730
is some number xi.

137
00:08:44,730 --> 00:08:47,600
We have a probabilistic
experiment where we pick one

138
00:08:47,600 --> 00:08:51,170
of the students at random, and
each student is equally likely

139
00:08:51,170 --> 00:08:53,820
to be picked as any
other student.

140
00:08:53,820 --> 00:08:56,650
And we're interested in the
random variable X, which is

141
00:08:56,650 --> 00:08:59,970
the weight of the student
that was selected.

142
00:08:59,970 --> 00:09:02,300
To keep things simple, we
will assume that the

143
00:09:02,300 --> 00:09:05,690
xi's are all distinct.

144
00:09:05,690 --> 00:09:11,540
And we first find the PMF
of this random variable.

145
00:09:11,540 --> 00:09:16,110
Any particular xi that this
possible is associated to

146
00:09:16,110 --> 00:09:18,620
exactly one student, because
we assumed that

147
00:09:18,620 --> 00:09:20,460
the xi's are distinct.

148
00:09:20,460 --> 00:09:24,140
So this probability would be the
probability or selecting

149
00:09:24,140 --> 00:09:29,770
the ith student, and that
probability is 1 over n.

150
00:09:29,770 --> 00:09:33,230
And now we can proceed and
calculate the expected value

151
00:09:33,230 --> 00:09:38,470
of the random variable X. This
random variable X takes

152
00:09:38,470 --> 00:09:43,340
values, and the values that
it takes are the xi's.

153
00:09:43,340 --> 00:09:48,540
A particular xi would be
associated with a probability

154
00:09:48,540 --> 00:09:53,840
1 over n, and we're adding over
all the i's or over all

155
00:09:53,840 --> 00:09:55,040
of the students.

156
00:09:55,040 --> 00:09:58,330
And so this is the
expected value.

157
00:09:58,330 --> 00:10:03,910
What we have here is just the
average of the weights of the

158
00:10:03,910 --> 00:10:06,060
students in this class.

159
00:10:06,060 --> 00:10:10,360
So the expected value in this
particular experiment can be

160
00:10:10,360 --> 00:10:14,550
interpreted as the true average
over the entire

161
00:10:14,550 --> 00:10:17,550
population of the students.

162
00:10:17,550 --> 00:10:19,400
Of course, here we're
talking about two

163
00:10:19,400 --> 00:10:21,730
different kinds of averages.

164
00:10:21,730 --> 00:10:25,280
In some sense, we're thinking
of expected values as the

165
00:10:25,280 --> 00:10:28,540
average in a large number of
repetitions of experiments.

166
00:10:28,540 --> 00:10:32,040
But here we have another
interpretation as the average

167
00:10:32,040 --> 00:10:34,040
over a particular population.