1
00:00:00,000 --> 00:00:07,310
PROFESSOR: Hi.

2
00:00:07,310 --> 00:00:08,970
Welcome back to recitation.

3
00:00:08,970 --> 00:00:10,970
Today we're going to do
a couple of exercises

4
00:00:10,970 --> 00:00:16,040
on computing average values
and probabilities using

5
00:00:16,040 --> 00:00:18,669
integration.

6
00:00:18,669 --> 00:00:20,210
So for this problem,
I'm going to let

7
00:00:20,210 --> 00:00:22,800
R be the name of
the region that's

8
00:00:22,800 --> 00:00:26,330
bounded above by the
curve, by the line I guess,

9
00:00:26,330 --> 00:00:30,090
y equals x and below by
the curve y equals x cubed.

10
00:00:30,090 --> 00:00:31,770
So I have a little
picture of R here.

11
00:00:31,770 --> 00:00:34,790
It, you know, lives all in
the first quadrant there.

12
00:00:34,790 --> 00:00:38,296
It's just this little sliver.

13
00:00:38,296 --> 00:00:39,670
So the first part
of the question

14
00:00:39,670 --> 00:00:43,270
is, if I look at
all the points in R,

15
00:00:43,270 --> 00:00:46,530
what's that average
x-coordinate of those points?

16
00:00:46,530 --> 00:00:49,220
What's the average value
of the x-coordinate?

17
00:00:49,220 --> 00:00:52,860
And the second problem is,
if I choose a point at random

18
00:00:52,860 --> 00:00:55,310
somewhere in R,
what's the probability

19
00:00:55,310 --> 00:00:58,050
that its x-coordinate
is larger than 1/2?

20
00:00:58,050 --> 00:00:59,850
So what's the
probability that it lies

21
00:00:59,850 --> 00:01:02,225
on the right-hand side of R?

22
00:01:02,225 --> 00:01:03,600
So why don't you
pause the video,

23
00:01:03,600 --> 00:01:05,960
take a couple minutes to
work through these problems,

24
00:01:05,960 --> 00:01:07,793
come back, and we can
work on them together.

25
00:01:07,793 --> 00:01:16,240


26
00:01:16,240 --> 00:01:17,520
Alright, welcome back.

27
00:01:17,520 --> 00:01:20,400
So hopefully you've had a chance
to get some good work done

28
00:01:20,400 --> 00:01:21,630
on these questions.

29
00:01:21,630 --> 00:01:23,800
So let's start to
work through them.

30
00:01:23,800 --> 00:01:29,290
So remember, to compute the
average value of a function

31
00:01:29,290 --> 00:01:31,420
over some region
what you need to do

32
00:01:31,420 --> 00:01:33,430
is you need to compute
a weighted average.

33
00:01:33,430 --> 00:01:37,910
And the weighting here is that
you have to consider the fact

34
00:01:37,910 --> 00:01:40,300
that, you know, for
x very near zero,

35
00:01:40,300 --> 00:01:43,520
there aren't very many points
in R in this little corner.

36
00:01:43,520 --> 00:01:44,020
Right?

37
00:01:44,020 --> 00:01:46,180
And for x very near
1, there aren't

38
00:01:46,180 --> 00:01:47,480
very many points there either.

39
00:01:47,480 --> 00:01:50,730
In the middle, this
region is a little higher,

40
00:01:50,730 --> 00:01:52,960
so there are more points there.

41
00:01:52,960 --> 00:01:54,910
So those points
will sort of weigh

42
00:01:54,910 --> 00:01:57,700
more when you take the
average of all points,

43
00:01:57,700 --> 00:01:59,110
than the points near the edges.

44
00:01:59,110 --> 00:02:02,040
So the way we account for
that is we have this weight

45
00:02:02,040 --> 00:02:04,450
function that is
the, in our case,

46
00:02:04,450 --> 00:02:06,960
since we're interested in
the x-coordinate, the weight

47
00:02:06,960 --> 00:02:10,310
at a given x-coordinate
is the slice

48
00:02:10,310 --> 00:02:14,770
of the-- how much of the region
lies above that x-coordinate.

49
00:02:14,770 --> 00:02:17,250
What, you know, the area of
a little rectangle above that

50
00:02:17,250 --> 00:02:19,640
x-coordinate says, tells
you how many of the points

51
00:02:19,640 --> 00:02:22,350
have that x-coordinate.

52
00:02:22,350 --> 00:02:27,135
So then we want to average
the function x, right?

53
00:02:27,135 --> 00:02:29,010
Because we're interested
in the x-coordinate,

54
00:02:29,010 --> 00:02:33,330
so the function
we're averaging is x.

55
00:02:33,330 --> 00:02:36,640
Over this region,
with that weighting.

56
00:02:36,640 --> 00:02:39,100
So, all right, so let's
write down what that means.

57
00:02:39,100 --> 00:02:52,150
So we have-- want average
of the function f of x,

58
00:02:52,150 --> 00:02:54,620
and f of x, the thing we're
computing the average of

59
00:02:54,620 --> 00:02:57,580
is, just the x-coordinate, so
it's just the function value x.

60
00:02:57,580 --> 00:03:00,130


61
00:03:00,130 --> 00:03:03,476
Over R. So we want the
average of this function, f

62
00:03:03,476 --> 00:03:06,730
of x, over R. So what
we need to compute

63
00:03:06,730 --> 00:03:09,980
is the-- so we have two
integrals we need to compute.

64
00:03:09,980 --> 00:03:13,260
We need one integral
that is the numerator,

65
00:03:13,260 --> 00:03:16,682
and so that numerator--
so I'm going

66
00:03:16,682 --> 00:03:20,180
to just write average for
the average that we want.

67
00:03:20,180 --> 00:03:23,400
So the numerator is
the integral-- OK,

68
00:03:23,400 --> 00:03:24,900
and so we have to
integrate, we have

69
00:03:24,900 --> 00:03:27,370
to take all possible
x-values into consideration.

70
00:03:27,370 --> 00:03:32,660
So x going, in this case
that's x going from 0 to 1

71
00:03:32,660 --> 00:03:35,640
and now we want to multiply the
function that we're averaging,

72
00:03:35,640 --> 00:03:38,779
which in this case is x, by the
appropriate weight function.

73
00:03:38,779 --> 00:03:40,820
And the weight function
is how much of the region

74
00:03:40,820 --> 00:03:43,060
is associated with that x-value.

75
00:03:43,060 --> 00:03:45,200
And that's the height of
this little rectangle,

76
00:03:45,200 --> 00:03:52,730
which in this case is
x minus x cubed dx.

77
00:03:52,730 --> 00:03:53,480
OK.

78
00:03:53,480 --> 00:03:55,460
But then, this is
an average, we have

79
00:03:55,460 --> 00:03:58,130
to divide by the total
weight of the region.

80
00:03:58,130 --> 00:04:01,190
The weight, in this
case, is just the area.

81
00:04:01,190 --> 00:04:03,390
So we have to divide
by the integral

82
00:04:03,390 --> 00:04:07,912
from 0 to 1 of just
this x minus x cubed dx.

83
00:04:07,912 --> 00:04:10,750


84
00:04:10,750 --> 00:04:12,670
OK, so we have to compute
these two integrals

85
00:04:12,670 --> 00:04:15,740
and then we have to
take their ratio.

86
00:04:15,740 --> 00:04:17,240
So let's do them separately.

87
00:04:17,240 --> 00:04:19,562
So the first one is--
well, the second one

88
00:04:19,562 --> 00:04:20,770
is actually a little simpler.

89
00:04:20,770 --> 00:04:24,190
The one in the denominator,
so let's handle that first.

90
00:04:24,190 --> 00:04:31,049
The integral from 0 to
1 of x minus x cubed dx.

91
00:04:31,049 --> 00:04:32,590
Well, this is a
pretty easy integral.

92
00:04:32,590 --> 00:04:38,380
It's x squared over 2 minus
x to the fourth over 4

93
00:04:38,380 --> 00:04:39,910
between 0 and 1.

94
00:04:39,910 --> 00:04:44,600
So that's 1/2 minus 1/4
minus, well when you put in 0

95
00:04:44,600 --> 00:04:45,630
you just get 0.

96
00:04:45,630 --> 00:04:48,870
So that's 1/4.

97
00:04:48,870 --> 00:04:52,300
And the first one,
the top, the numerator

98
00:04:52,300 --> 00:04:54,767
is the integral from 0 to 1.

99
00:04:54,767 --> 00:04:57,290
OK, we can multiply
through, so that's

100
00:04:57,290 --> 00:05:02,180
x squared minus x
to the fourth dx.

101
00:05:02,180 --> 00:05:05,030
So that integral
from 0 to 1 again

102
00:05:05,030 --> 00:05:14,980
is x cubed over 3 minus x to the
fifth over 5 between 0 and 1.

103
00:05:14,980 --> 00:05:19,270
When we put in 0 we get 0,
put in 1 we get 1/3 minus 1/5.

104
00:05:19,270 --> 00:05:23,220
So that's common denominator 15.

105
00:05:23,220 --> 00:05:25,360
2/15.

106
00:05:25,360 --> 00:05:29,040
So the numerator of our
average value is 2/15,

107
00:05:29,040 --> 00:05:32,610
the denominator is 1/4.

108
00:05:32,610 --> 00:05:34,680
So the average value
we're interested in

109
00:05:34,680 --> 00:05:41,770
is 2/15 divided by
1/4, which is 8/15.

110
00:05:41,770 --> 00:05:43,830
So the average
x-coordinate is just,

111
00:05:43,830 --> 00:05:47,290
so 8/15 is just a tiny
bit larger than 1/2.

112
00:05:47,290 --> 00:05:50,760
So this is saying somehow the
average x-coordinate is just

113
00:05:50,760 --> 00:05:53,840
slightly shifted to
the right of 1/2.

114
00:05:53,840 --> 00:05:56,630
So this, I may have drawn this
region sort of symmetrically,

115
00:05:56,630 --> 00:06:00,530
but in fact it's a little bit
shifted to the right there.

116
00:06:00,530 --> 00:06:03,500
So that's the the first
part of the problem.

117
00:06:03,500 --> 00:06:06,680


118
00:06:06,680 --> 00:06:11,146
For the second part, we want
to compute the probability-- so

119
00:06:11,146 --> 00:06:14,420
OK, so we choose a point
at random in this set R

120
00:06:14,420 --> 00:06:16,510
and we want to know
what's the probability

121
00:06:16,510 --> 00:06:20,530
that its x-coordinate
is larger than 1/2.

122
00:06:20,530 --> 00:06:22,940
Well, since all points are,
you know, equally likely,

123
00:06:22,940 --> 00:06:26,197
all regions, the
probability just

124
00:06:26,197 --> 00:06:27,780
has to do with the
area of the region.

125
00:06:27,780 --> 00:06:31,760
What we really want to know
is, what's is area of R

126
00:06:31,760 --> 00:06:34,960
to the right of the
line x equals 1/2.

127
00:06:34,960 --> 00:06:37,650
So that's the good
region, and then we

128
00:06:37,650 --> 00:06:40,330
want to know how much of
the entire region is that.

129
00:06:40,330 --> 00:06:43,300
So we want to know the
area of the good region

130
00:06:43,300 --> 00:06:47,014
to the right of 1/2
divided by the total area.

131
00:06:47,014 --> 00:06:48,430
Now luckily we've
already computed

132
00:06:48,430 --> 00:06:50,590
the total area, right here.

133
00:06:50,590 --> 00:06:53,610
This was this denominator
that we had over here.

134
00:06:53,610 --> 00:06:56,890
So we just need the
numerator of that fraction.

135
00:06:56,890 --> 00:07:00,000
So let's go over here,
let me write that down.

136
00:07:00,000 --> 00:07:04,300


137
00:07:04,300 --> 00:07:12,260
So the probability
equals-- I'm going

138
00:07:12,260 --> 00:07:22,710
to put good area in
quotation marks--

139
00:07:22,710 --> 00:07:26,330
so what I mean by good area is
the area of the set of points

140
00:07:26,330 --> 00:07:28,450
that satisfies our condition.

141
00:07:28,450 --> 00:07:35,240
And in our case, the
good area is just-- so

142
00:07:35,240 --> 00:07:38,380
a point we're interested in,
if its x-coordinate is bigger

143
00:07:38,380 --> 00:07:42,800
than 1/2, so we just want to
take the part of this region

144
00:07:42,800 --> 00:07:45,150
to the right of 1/2.

145
00:07:45,150 --> 00:07:48,410
So in order to
compute that, we just

146
00:07:48,410 --> 00:07:51,999
take this integral from 1/2
to 1 instead of from 0 to 1.

147
00:07:51,999 --> 00:07:54,290
So we're only counting those
points to the right of 1/2

148
00:07:54,290 --> 00:07:58,870
and then we want the,
you know, the area

149
00:07:58,870 --> 00:08:02,600
between these two
curves on that region.

150
00:08:02,600 --> 00:08:06,300
OK, so, and again this
is a fairly simple

151
00:08:06,300 --> 00:08:07,230
integral to compute.

152
00:08:07,230 --> 00:08:13,780
So this gives me x squared over
2 minus x to the fourth over 4

153
00:08:13,780 --> 00:08:16,390
between 1/2 and 1.

154
00:08:16,390 --> 00:08:18,850
All right, so this is
maybe a tiny bit hairy,

155
00:08:18,850 --> 00:08:26,560
so this gives me 1/2 minus 1/4
minus-- well we put in 1/2 here

156
00:08:26,560 --> 00:08:35,260
we get 1/8 minus-- 1/2 to the
fourth is 1/16, divided by 4

157
00:08:35,260 --> 00:08:36,100
is 1/64.

158
00:08:36,100 --> 00:08:39,050


159
00:08:39,050 --> 00:08:42,861
So this is all going
to be in sixty-fourths.

160
00:08:42,861 --> 00:08:43,360
So OK.

161
00:08:43,360 --> 00:08:51,700
So let's see, 32/64 minus
16/64 minus 8/64 plus 1.

162
00:08:51,700 --> 00:08:54,075
If I just put it all over
that common denominator.

163
00:08:54,075 --> 00:08:56,930


164
00:08:56,930 --> 00:09:03,022
All right, so I think that looks
like 9/64, if I did that right.

165
00:09:03,022 --> 00:09:03,522
So, OK.

166
00:09:03,522 --> 00:09:04,920
So that's the good area.

167
00:09:04,920 --> 00:09:06,540
And then the
probability that I want,

168
00:09:06,540 --> 00:09:09,670
I have to divide the good
area by the total area.

169
00:09:09,670 --> 00:09:13,540
And we saw before that
the total area was 1/4.

170
00:09:13,540 --> 00:09:16,870
And that was what
this computation was.

171
00:09:16,870 --> 00:09:20,990
So the total area's 1/4,
so the probability--

172
00:09:20,990 --> 00:09:23,840
I'm just going to write pr
for probability-- that we're

173
00:09:23,840 --> 00:09:34,190
interested in is 9/64
divided by 1/4 which is 9/16.

174
00:09:34,190 --> 00:09:37,340


175
00:09:37,340 --> 00:09:41,020
OK, so this says-- also 9/16
is also a little bit bigger

176
00:09:41,020 --> 00:09:41,600
than a half.

177
00:09:41,600 --> 00:09:44,810
So this is a different way of
saying, our region, there's

178
00:09:44,810 --> 00:09:46,410
a little bit more
of it to the right

179
00:09:46,410 --> 00:09:47,987
then there is to the left.

180
00:09:47,987 --> 00:09:49,570
You know, so it's
slightly more likely

181
00:09:49,570 --> 00:09:52,130
that a random point is
to the right of the line

182
00:09:52,130 --> 00:09:55,940
y equals 1/2 than it is
to the left of that line.

183
00:09:55,940 --> 00:09:59,560
So just to sum up, we had
these two different problems

184
00:09:59,560 --> 00:10:01,520
that we did concerning
this region R.

185
00:10:01,520 --> 00:10:05,490
So first we computed the average
value of the x-coordinate

186
00:10:05,490 --> 00:10:06,850
of a point in this region.

187
00:10:06,850 --> 00:10:09,539


188
00:10:09,539 --> 00:10:10,330
So that was part a.

189
00:10:10,330 --> 00:10:12,592
We computed the average
value of the x-coordinate,

190
00:10:12,592 --> 00:10:13,550
and that was over here.

191
00:10:13,550 --> 00:10:16,230
So we had to, you know, do
this, the weighted average

192
00:10:16,230 --> 00:10:19,240
of the x-coordinate
divided by the total area.

193
00:10:19,240 --> 00:10:21,750
And then second, we
computed the probability

194
00:10:21,750 --> 00:10:25,760
that a randomly chosen point has
x-coordinate larger than 1/2.

195
00:10:25,760 --> 00:10:28,300
And we did that over here.

196
00:10:28,300 --> 00:10:29,930
And for that we
needed to compute

197
00:10:29,930 --> 00:10:32,290
the area of the
good region, which

198
00:10:32,290 --> 00:10:35,090
was the part to the right
of the line y equals 1/2

199
00:10:35,090 --> 00:10:37,820
and then we needed to
divide it by the total area.

200
00:10:37,820 --> 00:10:39,139
So I'll end there.

201
00:10:39,139 --> 00:10:39,639