1
00:00:00,820 --> 00:00:03,680
In this segment, we discuss the
expected value rule for

2
00:00:03,680 --> 00:00:05,890
calculating the expected
value of a

3
00:00:05,890 --> 00:00:08,000
function of a random variable.

4
00:00:08,000 --> 00:00:11,180
It corresponds to a nice formula
that we will see

5
00:00:11,180 --> 00:00:15,220
shortly, but it also involves a
much more general idea that

6
00:00:15,220 --> 00:00:18,970
we will encounter many times
in this course, in somewhat

7
00:00:18,970 --> 00:00:20,480
different forms.

8
00:00:20,480 --> 00:00:22,920
Here's what it is all about.

9
00:00:22,920 --> 00:00:27,910
We start with a certain random
variable that has a known PMF.

10
00:00:27,910 --> 00:00:30,720
However, we're ultimately
interested in another random

11
00:00:30,720 --> 00:00:34,190
variable Y, which is defined as
a function of the original

12
00:00:34,190 --> 00:00:35,460
random variable.

13
00:00:35,460 --> 00:00:38,130
We're interested in calculating
the expected value

14
00:00:38,130 --> 00:00:42,130
of this new random variable,
Y. How should we do it?

15
00:00:42,130 --> 00:00:45,150
We will illustrate the ideas
involved through a simple

16
00:00:45,150 --> 00:00:46,670
numerical example.

17
00:00:46,670 --> 00:00:49,810
In this example, we have a
random variable, X, that takes

18
00:00:49,810 --> 00:00:52,910
values 2, 3, 4, or 5, according
to some given

19
00:00:52,910 --> 00:00:54,140
probabilities.

20
00:00:54,140 --> 00:00:56,570
We are also given a
function that maps

21
00:00:56,570 --> 00:00:59,300
x-values into y-values.

22
00:00:59,300 --> 00:01:04,080
And this function, g, then
defines a new random variable.

23
00:01:04,080 --> 00:01:07,550
So if the outcome of the
experiment leads to an X equal

24
00:01:07,550 --> 00:01:11,200
to 4, then the random variable,
Y, will also take a

25
00:01:11,200 --> 00:01:13,130
value equal to 4.

26
00:01:13,130 --> 00:01:16,200
How do we calculate the
expected value of Y?

27
00:01:16,200 --> 00:01:18,720
The only tool that we have
available in our hands at this

28
00:01:18,720 --> 00:01:22,310
point is the definition of the
expected value, which tells us

29
00:01:22,310 --> 00:01:26,620
that we should run a summation
over the y-axis, consider

30
00:01:26,620 --> 00:01:29,520
different values of
y one at the time.

31
00:01:29,520 --> 00:01:33,640
And for each value for y,
multiply that value by its

32
00:01:33,640 --> 00:01:35,430
corresponding probability.

33
00:01:35,430 --> 00:01:39,640
So in this case, we start with
Y equal to 3, which needs to

34
00:01:39,640 --> 00:01:41,870
be multiplied by the
probability that

35
00:01:41,870 --> 00:01:43,620
Y is equal to 3.

36
00:01:43,620 --> 00:01:45,150
What is that probability?

37
00:01:45,150 --> 00:01:49,850
Well, Y is equal to three, if
and only if X is 2 or 3, which

38
00:01:49,850 --> 00:01:56,430
happens with probability,
0.1 plus 0.2.

39
00:01:56,430 --> 00:01:59,400
Then we continue with the
summation by considering the

40
00:01:59,400 --> 00:02:01,790
next value of little y.

41
00:02:01,790 --> 00:02:04,280
The next possible value is 4.

42
00:02:04,280 --> 00:02:08,009
And this gives us a contribution
of 4, weighted by

43
00:02:08,009 --> 00:02:10,490
the probability of
obtaining a 4.

44
00:02:10,490 --> 00:02:13,340
The probability that Y is equal
to 4 is the probability

45
00:02:13,340 --> 00:02:17,579
that X is either equal to 4 or
to 5, which happens with

46
00:02:17,579 --> 00:02:18,340
probability.

47
00:02:18,340 --> 00:02:20,750
0.3 plus 0.4.

48
00:02:20,750 --> 00:02:24,660
So this way, we obtain an
arithmetic expression which we

49
00:02:24,660 --> 00:02:25,890
can evaluate.

50
00:02:25,890 --> 00:02:29,480
And it's going to give us the
expected value of Y. But

51
00:02:29,480 --> 00:02:31,680
here's an alternative
way of calculating

52
00:02:31,680 --> 00:02:33,400
the expected value.

53
00:02:33,400 --> 00:02:38,310
And this corresponds to the
following type of thinking.

54
00:02:38,310 --> 00:02:42,260
10% of the time, X is going
to be equal to 2.

55
00:02:42,260 --> 00:02:46,079
And when that happens, Y
takes on a value of 3.

56
00:02:46,079 --> 00:02:49,930
So this should give us a
contribution to the average

57
00:02:49,930 --> 00:02:55,430
value of Y, which
is 3 times 0.1.

58
00:02:55,430 --> 00:03:02,590
Then, 20% of the time, X
is 3 and Y is also 3.

59
00:03:02,590 --> 00:03:09,980
So 20% of the time, we
also get 3's in Y.

60
00:03:09,980 --> 00:03:15,790
Then 30% of the time, X is 4,
which results in a Y that's

61
00:03:15,790 --> 00:03:17,280
equal to 4.

62
00:03:17,280 --> 00:03:21,620
So we obtain a 4 30%
of the time.

63
00:03:21,620 --> 00:03:26,440
And finally, 40% of the time, X
equals to [5], which results

64
00:03:26,440 --> 00:03:30,329
in a Y equal to 4.

65
00:03:30,329 --> 00:03:33,710
And we obtain this arithmetic
expression.

66
00:03:33,710 --> 00:03:36,600
Now you can compare the two
arithmetic expressions, the

67
00:03:36,600 --> 00:03:39,940
red and the blue one, and you
will notice that they're

68
00:03:39,940 --> 00:03:43,470
equal, except that the terms
are arranged in a slightly

69
00:03:43,470 --> 00:03:44,880
different way.

70
00:03:44,880 --> 00:03:48,210
Conceptually, however, there's
a very big difference.

71
00:03:48,210 --> 00:03:50,920
In the first summation, we
run over the values of

72
00:03:50,920 --> 00:03:53,079
Y one at the time.

73
00:03:53,079 --> 00:03:56,630
In the second summation, we run
over the different values

74
00:03:56,630 --> 00:04:01,320
of X one at a time, and took
into account their individual

75
00:04:01,320 --> 00:04:02,990
contributions.

76
00:04:02,990 --> 00:04:07,500
This second way of calculating
the expected value of Y is

77
00:04:07,500 --> 00:04:09,710
called the expected
value rule.

78
00:04:09,710 --> 00:04:13,560
And it corresponds to the
following formula.

79
00:04:13,560 --> 00:04:17,390
We carry out a summation
over the x-axis.

80
00:04:17,390 --> 00:04:22,330
For each x-value that we
consider, we calculate what is

81
00:04:22,330 --> 00:04:27,830
the corresponding y-value,
that's g of x, and also weigh

82
00:04:27,830 --> 00:04:30,620
this term according to
the probability of

83
00:04:30,620 --> 00:04:32,510
this particular x.

84
00:04:32,510 --> 00:04:37,710
So for instance, a typical term
here would be when x is

85
00:04:37,710 --> 00:04:42,440
equal to 2, g of x would
be equal to 3.

86
00:04:42,440 --> 00:04:44,480
And the corresponding
probability, that's the

87
00:04:44,480 --> 00:04:49,560
probability of a 2,
would be 0.1.

88
00:04:49,560 --> 00:04:52,790
The advantage of using the
expected value rule instead of

89
00:04:52,790 --> 00:04:55,180
the definition of the
expectation is that the

90
00:04:55,180 --> 00:04:58,510
expected value rule only
involves the PMF of the

91
00:04:58,510 --> 00:05:02,540
original random variable, so
we do not need to do any

92
00:05:02,540 --> 00:05:05,295
additional work to
find the PMF of

93
00:05:05,295 --> 00:05:08,230
the new random variable.

94
00:05:08,230 --> 00:05:13,610
Now we argued in favor of the
expected value rule by

95
00:05:13,610 --> 00:05:16,450
considering this numerical
example, and by checking that

96
00:05:16,450 --> 00:05:18,730
it gives the right result.

97
00:05:18,730 --> 00:05:20,520
But now let us verify.

98
00:05:20,520 --> 00:05:23,760
Let us argue more generally that
it's going to give us the

99
00:05:23,760 --> 00:05:25,410
right answer.

100
00:05:25,410 --> 00:05:30,300
So what we're going to do is
to take this summation and

101
00:05:30,300 --> 00:05:34,980
argue that it's equal to the
expected value of Y, which is

102
00:05:34,980 --> 00:05:37,040
defined by that summation.

103
00:05:37,040 --> 00:05:38,510
So let us start with this.

104
00:05:38,510 --> 00:05:41,790
It's a sum over all x's.

105
00:05:41,790 --> 00:05:48,920
Let us first fix a particular
value of y, and add over all

106
00:05:48,920 --> 00:05:52,970
those x's that correspond
to that particular y.

107
00:05:52,970 --> 00:05:56,440
So we're fixing a
particular y.

108
00:05:56,440 --> 00:06:02,320
And so we're adding only over
those x's that lead to that

109
00:06:02,320 --> 00:06:04,170
particular y.

110
00:06:04,170 --> 00:06:06,085
And we carry out to
the summation.

111
00:06:09,680 --> 00:06:13,500
So this is the part of this
sum associated with one

112
00:06:13,500 --> 00:06:16,620
particular choice of y.

113
00:06:16,620 --> 00:06:20,000
And it's a sum, really,
over this set of x's.

114
00:06:20,000 --> 00:06:24,370
But in order to exhaust all x's,
we need to consider all

115
00:06:24,370 --> 00:06:26,930
possible values of y.

116
00:06:26,930 --> 00:06:31,690
And this gives rise to an
outer summation over the

117
00:06:31,690 --> 00:06:33,130
different y's.

118
00:06:33,130 --> 00:06:37,630
So for any fixed y, we add
over the associated x's.

119
00:06:37,630 --> 00:06:40,250
But we want to consider
all the possible y's.

120
00:06:43,010 --> 00:06:47,960
Now at this point, we make the
following observation.

121
00:06:50,460 --> 00:06:53,460
Here, we have a summation
over y's.

122
00:06:53,460 --> 00:06:57,710
And let's look at the
inner summation.

123
00:06:57,710 --> 00:07:03,520
The inner summation involves
x's, all of which are

124
00:07:03,520 --> 00:07:07,140
associated with a specific
value of y.

125
00:07:07,140 --> 00:07:13,360
Having fixed y, all the terms
inside this sum have the

126
00:07:13,360 --> 00:07:16,290
property that g of
x is equal to y.

127
00:07:16,290 --> 00:07:20,960
So g of x is equal to
that particular y.

128
00:07:20,960 --> 00:07:25,710
And we can make this
substitution here.

129
00:07:25,710 --> 00:07:28,510
Now when we look at this
summation, we now realize that

130
00:07:28,510 --> 00:07:33,470
it's a summation over x's
while y is being fixed.

131
00:07:33,470 --> 00:07:37,690
And so we can take this
term of y and pull

132
00:07:37,690 --> 00:07:39,165
it outside the summation.

133
00:07:42,620 --> 00:07:50,590
What this leaves us with is a
sum over all y's of y, and

134
00:07:50,590 --> 00:07:54,930
then a further sum over all
x's that lead to that

135
00:07:54,930 --> 00:07:59,590
particular y, of the
probabilities of those x's.

136
00:08:04,640 --> 00:08:09,850
Now what can we say about
this inner summation?

137
00:08:09,850 --> 00:08:12,960
We have fixed a y.

138
00:08:12,960 --> 00:08:16,050
For that particular y, we're
adding the probabilities of

139
00:08:16,050 --> 00:08:19,910
all the x's that lead to
that particular y.

140
00:08:19,910 --> 00:08:23,820
Fixing y, consider all the
x's that lead to it.

141
00:08:23,820 --> 00:08:27,100
This is just the probability
of that particular y.

142
00:08:29,740 --> 00:08:33,549
But what we have now is just the
definition of the expected

143
00:08:33,549 --> 00:08:38,840
value of Y. And this concludes
the proof that this

144
00:08:38,840 --> 00:08:42,490
expression, as given by the
expected value rule, gives us

145
00:08:42,490 --> 00:08:45,530
the same answer as the original
definition of the

146
00:08:45,530 --> 00:08:48,110
expected value of Y.

147
00:08:48,110 --> 00:08:50,860
Now before closing, a
few observations.

148
00:08:50,860 --> 00:08:54,350
The expected value rule is
really simple to use.

149
00:08:54,350 --> 00:08:56,890
For example, if you want to
calculate the expected value

150
00:08:56,890 --> 00:09:00,330
of the square of a random
variable, then you're dealing

151
00:09:00,330 --> 00:09:03,560
with a situation where
the g function

152
00:09:03,560 --> 00:09:07,350
is the square function.

153
00:09:07,350 --> 00:09:11,790
And so, the expected value of
X-squared will be the sum over

154
00:09:11,790 --> 00:09:16,700
x's of x squared weighted
according to the probability

155
00:09:16,700 --> 00:09:19,260
of a particular x.

156
00:09:19,260 --> 00:09:22,950
And finally, one important
word of caution, that in

157
00:09:22,950 --> 00:09:26,090
general, the expected value
of the function--

158
00:09:26,090 --> 00:09:29,750
so for example, the expected
value of X-squared.

159
00:09:29,750 --> 00:09:35,480
In general, it's not going to
be the same as taking the

160
00:09:35,480 --> 00:09:40,620
expected value of X
and squaring it.

161
00:09:40,620 --> 00:09:43,800
So this is a word [of] caution,
that in general, you

162
00:09:43,800 --> 00:09:46,960
cannot interchange the order
with which you apply a

163
00:09:46,960 --> 00:09:50,820
function, and then you calculate
expectation.

164
00:09:50,820 --> 00:09:54,060
There are exceptions, however,
in which we happen to have

165
00:09:54,060 --> 00:09:55,400
equality here.

166
00:09:55,400 --> 00:09:57,320
And this is going to
be our next topic.