1
00:00:01,080 --> 00:00:03,660
We have claimed that normal
random variables are very

2
00:00:03,660 --> 00:00:06,880
important, and therefore we
would like to be able to

3
00:00:06,880 --> 00:00:09,280
calculate probabilities
associated with them.

4
00:00:09,280 --> 00:00:11,940
For example, given a normal
random variable, what is the

5
00:00:11,940 --> 00:00:15,200
probability that it takes
a value less than 5?

6
00:00:15,200 --> 00:00:18,470
Unfortunately, there are no
closed form expressions that

7
00:00:18,470 --> 00:00:20,210
can help us with this.

8
00:00:20,210 --> 00:00:23,040
In particular, the CDF, the
Cumulative Distribution

9
00:00:23,040 --> 00:00:26,030
Function of normal random
variables, is not given in

10
00:00:26,030 --> 00:00:27,120
closed form.

11
00:00:27,120 --> 00:00:30,960
But fortunately, we do have
tables for the standard normal

12
00:00:30,960 --> 00:00:32,820
random variable.

13
00:00:32,820 --> 00:00:38,270
These tables, which take the
form shown here, give us the

14
00:00:38,270 --> 00:00:40,200
following information.

15
00:00:40,200 --> 00:00:43,930
If we have a normal random
variable, which is a standard

16
00:00:43,930 --> 00:00:48,150
normal, they tell us the values
of the cumulative

17
00:00:48,150 --> 00:00:55,240
distribution function for
different values of little y.

18
00:00:55,240 --> 00:00:59,870
In terms of a picture, if this
is the PDF of a standard

19
00:00:59,870 --> 00:01:04,840
normal and I give you a value
little y, I'm interested in

20
00:01:04,840 --> 00:01:08,250
the corresponding value of
the CDF, which is the

21
00:01:08,250 --> 00:01:10,150
area under the curve.

22
00:01:10,150 --> 00:01:13,770
Well, that value, the area under
this curve, is exactly

23
00:01:13,770 --> 00:01:16,580
what this table is
giving to us.

24
00:01:16,580 --> 00:01:19,990
And there's a shorthand notation
for referring to the

25
00:01:19,990 --> 00:01:24,510
CDF of the standard normal,
which is just phi of y.

26
00:01:24,510 --> 00:01:27,390
Let us see how we
use this table.

27
00:01:27,390 --> 00:01:31,140
Suppose we're interested
in phi of 0.

28
00:01:31,140 --> 00:01:33,900
Which is the probability that
our standard normal takes a

29
00:01:33,900 --> 00:01:36,570
value less than or equal to 0?

30
00:01:36,570 --> 00:01:40,259
Well, by symmetry since the PDF
is symmetric around 0, we

31
00:01:40,259 --> 00:01:44,180
know that this probability
should be 0.5.

32
00:01:44,180 --> 00:01:46,990
Let's see what the
table tells us.

33
00:01:46,990 --> 00:01:53,690
0 corresponds to this entry,
which is indeed 0.5.

34
00:01:53,690 --> 00:01:57,789
Let us look up the probability
that our standard normal takes

35
00:01:57,789 --> 00:02:01,330
a value less than,
let's say, 1.16.

36
00:02:01,330 --> 00:02:03,580
How do we find this
information?

37
00:02:03,580 --> 00:02:06,620
1 is here.

38
00:02:06,620 --> 00:02:11,770
And 1.1 is here.

39
00:02:11,770 --> 00:02:17,350
1.1, and then we have a 6 in the
next decimal place, which

40
00:02:17,350 --> 00:02:20,590
leads us to this entry.

41
00:02:20,590 --> 00:02:27,660
And so this value is 0.8770.

42
00:02:27,660 --> 00:02:31,200
Similarly, we can calculate
the probability that the

43
00:02:31,200 --> 00:02:35,030
normal is less than 2.9.

44
00:02:35,030 --> 00:02:37,010
How do we look up this
information?

45
00:02:37,010 --> 00:02:38,860
2.9 is here.

46
00:02:38,860 --> 00:02:42,520
We do not have another decimal
digit, so we're looking at

47
00:02:42,520 --> 00:02:43,640
this column.

48
00:02:43,640 --> 00:02:50,910
And we obtain this value,
which is 0.9981.

49
00:02:50,910 --> 00:02:54,930
And by looking at this number
we can actually tell that a

50
00:02:54,930 --> 00:02:59,810
standard normal random variable
has extremely low

51
00:02:59,810 --> 00:03:04,400
probability of being
bigger than 2.9.

52
00:03:04,400 --> 00:03:08,850
Now notice that the table
specifies phi of y for y being

53
00:03:08,850 --> 00:03:10,190
non-negative.

54
00:03:10,190 --> 00:03:15,820
What if we wish to calculate the
value, for example, of phi

55
00:03:15,820 --> 00:03:18,860
of minus 2?

56
00:03:18,860 --> 00:03:23,980
In terms of a picture, this
is a standard normal.

57
00:03:23,980 --> 00:03:26,870
Here is minus 2.

58
00:03:26,870 --> 00:03:29,570
And we wish to calculate
this probability.

59
00:03:29,570 --> 00:03:32,040
There's nothing in the table
that gives us this probability

60
00:03:32,040 --> 00:03:35,150
directly, but we can
argue as follows.

61
00:03:35,150 --> 00:03:37,730
The normal PDF is symmetric.

62
00:03:37,730 --> 00:03:43,530
So if we look at 2, then this
probability here, which is phi

63
00:03:43,530 --> 00:03:47,200
of minus 2, is the same
as that probability

64
00:03:47,200 --> 00:03:49,020
here, of that tail.

65
00:03:49,020 --> 00:03:51,810
What is the probability
of that tail?

66
00:03:51,810 --> 00:03:56,750
It's 1, which is the overall
area under the curve, minus

67
00:03:56,750 --> 00:04:01,530
the area under the curve when
you go up to the value of 2.

68
00:04:04,540 --> 00:04:11,340
So this quantity is going to be
the same as phi of minus 2.

69
00:04:11,340 --> 00:04:14,570
And this one we can now
get from the tables.

70
00:04:14,570 --> 00:04:15,830
It's 1 minus--

71
00:04:15,830 --> 00:04:18,709
let us see, 2 is here.

72
00:04:18,709 --> 00:04:27,170
It's 1 minus 0.9772.

73
00:04:27,170 --> 00:04:30,460
The standard normal table
gives us probabilities

74
00:04:30,460 --> 00:04:33,610
associated with a standard
normal random variable.

75
00:04:33,610 --> 00:04:37,320
What if we're dealing with a
normal random variable that

76
00:04:37,320 --> 00:04:41,450
has a mean and a variance that
are different from those of

77
00:04:41,450 --> 00:04:42,820
the standard normal?

78
00:04:42,820 --> 00:04:44,280
What can we do?

79
00:04:44,280 --> 00:04:48,200
Well, there's a general trick
that you can do to a random

80
00:04:48,200 --> 00:04:51,409
variable, which is
the following.

81
00:04:51,409 --> 00:04:55,770
Let us define a new random
variable Y in this fashion.

82
00:04:55,770 --> 00:05:01,030
Y measures how far away is
X from the mean value.

83
00:05:01,030 --> 00:05:04,610
But because we divide by sigma,
the standard deviation,

84
00:05:04,610 --> 00:05:08,920
it measures this distance
in standard deviations.

85
00:05:08,920 --> 00:05:14,190
So if Y is equal to 3 it means
that X is 3 standard

86
00:05:14,190 --> 00:05:16,320
deviations away from the mean.

87
00:05:16,320 --> 00:05:20,540
In general, Y measures how many
deviations away from the

88
00:05:20,540 --> 00:05:22,270
mean are you.

89
00:05:22,270 --> 00:05:25,030
What properties does this
random variable have?

90
00:05:25,030 --> 00:05:30,080
The expected value of Y is
going to be equal to 0,

91
00:05:30,080 --> 00:05:33,650
because we have X and we're
subtracting the mean of X. So

92
00:05:33,650 --> 00:05:36,890
the expected value of this
term is equal to 0.

93
00:05:36,890 --> 00:05:39,690
How about the variance of Y?

94
00:05:39,690 --> 00:05:43,820
Whenever we multiply a random
variable by a constant, the

95
00:05:43,820 --> 00:05:48,300
variance gets multiplied by the
square of that constant.

96
00:05:48,300 --> 00:05:51,320
So we get this expression.

97
00:05:51,320 --> 00:05:54,280
But the variance of X
is sigma squared.

98
00:05:54,280 --> 00:05:55,980
So this is equal to 1.

99
00:05:55,980 --> 00:05:59,500
So starting from X, we have
obtained a closely related

100
00:05:59,500 --> 00:06:03,460
random variable Y that has the
property that it has 0 mean

101
00:06:03,460 --> 00:06:05,480
and unit variance.

102
00:06:05,480 --> 00:06:10,990
If it also happens that X is a
normal random variable, then Y

103
00:06:10,990 --> 00:06:14,380
is going to be a standard
normal random variable.

104
00:06:14,380 --> 00:06:18,160
So we have managed to relate
X to a standard

105
00:06:18,160 --> 00:06:19,940
normal random variable.

106
00:06:19,940 --> 00:06:23,400
And perhaps you can rewrite this
expression in this form,

107
00:06:23,400 --> 00:06:29,040
X equals to mu plus
sigma Y where Y is

108
00:06:29,040 --> 00:06:32,620
now a standard normal.

109
00:06:32,620 --> 00:06:35,040
So, instead of doing
calculations having to do with

110
00:06:35,040 --> 00:06:40,220
X, we can try to calculate in
terms of Y. And for Y we do

111
00:06:40,220 --> 00:06:42,570
have available tables.

112
00:06:42,570 --> 00:06:46,300
Let us look at an example
of how this is done.

113
00:06:46,300 --> 00:06:49,210
The way to calculate
probabilities associated with

114
00:06:49,210 --> 00:06:53,560
general normal random variables
is to take the event

115
00:06:53,560 --> 00:06:58,140
whose probability we want
calculated and express it in

116
00:06:58,140 --> 00:07:00,690
terms of standard normal
random variables.

117
00:07:00,690 --> 00:07:03,810
And then use the standard
normal tables.

118
00:07:03,810 --> 00:07:07,090
Let us see how this is done
in terms of an example.

119
00:07:07,090 --> 00:07:13,000
Suppose that X is normal with
mean 6 and variance 4, so that

120
00:07:13,000 --> 00:07:15,780
the standard deviation
sigma is equal to 2.

121
00:07:15,780 --> 00:07:19,080
And suppose that we want to
calculate the probability that

122
00:07:19,080 --> 00:07:23,550
X lies between 2 and 8.

123
00:07:30,520 --> 00:07:33,200
Here's how we can proceed.

124
00:07:33,200 --> 00:07:39,140
This event is the same as the
event that X minus 6 takes a

125
00:07:39,140 --> 00:07:43,220
value between 2 minus
6 and 8 minus 6.

126
00:07:43,220 --> 00:07:45,790
This event is the same as the
original event we were

127
00:07:45,790 --> 00:07:47,590
interested in.

128
00:07:47,590 --> 00:07:50,890
We can also divide both sides
of this inequality by the

129
00:07:50,890 --> 00:07:52,420
standard deviation.

130
00:07:52,420 --> 00:07:55,040
And the event of interest
has now been

131
00:07:55,040 --> 00:07:59,430
expressed in this form.

132
00:07:59,430 --> 00:08:03,830
But at this point we recognize
that this is of the form X

133
00:08:03,830 --> 00:08:06,390
minus mu over sigma.

134
00:08:06,390 --> 00:08:09,430
So this random variable
here is a standard

135
00:08:09,430 --> 00:08:10,855
normal random variable.

136
00:08:14,090 --> 00:08:24,530
So the probability that X lies
between 2 and 8 is the same as

137
00:08:24,530 --> 00:08:27,600
the probability that a standard
normal random

138
00:08:27,600 --> 00:08:32,350
variable, call it Y, falls
between these numbers minus 4

139
00:08:32,350 --> 00:08:35,090
divided by 2, that's minus 2.

140
00:08:35,090 --> 00:08:37,380
Then Y less than 1.

141
00:08:40,960 --> 00:08:44,340
And now we can use the standard
normal tables to

142
00:08:44,340 --> 00:08:46,210
calculate this probability.

143
00:08:46,210 --> 00:08:49,960
We have here 1 and here
we have minus 2.

144
00:08:49,960 --> 00:08:52,790
And we want to find the
probability that our standard

145
00:08:52,790 --> 00:08:55,660
normal falls inside
this range.

146
00:08:55,660 --> 00:08:59,420
This is the probability that
it is less than 1.

147
00:08:59,420 --> 00:09:02,640
But we need to subtract the
probability of that tail so

148
00:09:02,640 --> 00:09:06,850
that we're left just with
this intermediate area.

149
00:09:06,850 --> 00:09:11,970
So this is the probability that
Y is less than 1 minus

150
00:09:11,970 --> 00:09:17,390
the probability that Y
is less than minus 2.

151
00:09:17,390 --> 00:09:21,520
And finally, as we discussed
earlier, the probability that

152
00:09:21,520 --> 00:09:27,360
Y is less than minus 2, this
is 1 minus the probability

153
00:09:27,360 --> 00:09:31,690
that Y is less than
or equal to 2.

154
00:09:31,690 --> 00:09:35,670
And now we can go to the normal
tables, identify the

155
00:09:35,670 --> 00:09:38,660
values that we're interested in,
the probability that Y is

156
00:09:38,660 --> 00:09:41,970
less than 1, the probability
that Y is less than 2, and

157
00:09:41,970 --> 00:09:43,440
plug these in.

158
00:09:43,440 --> 00:09:46,250
And this gives us the
desired probability.

159
00:09:46,250 --> 00:09:53,450
Again, the key step is to take
the event of interest and by

160
00:09:53,450 --> 00:09:55,530
subtracting the mean and
dividing by the standard

161
00:09:55,530 --> 00:09:59,740
deviation express that same
event in an equivalent form,

162
00:09:59,740 --> 00:10:02,240
but which now involves
a standard

163
00:10:02,240 --> 00:10:03,660
normal random variable.

164
00:10:06,880 --> 00:10:09,780
And then finally, use the
standard normal tables.