EC 271
Applied Statistical Methods: Lecture Notes (Leshui He)
2014
3
Chapter 6: Normal Probability Distribution
3.1
Primer
3.1.1
Probability (Readings in Chapter 4)
Defnition.
The
probability
is the chance that a particular event will occur. We denote the
probability of outcome
E
by
P
(
E
)
.
For example,
P
(
ad
iceturnedouttobe3
)
=
1
°
6.
Remark.
Three Important Properties of Probability
1.
0
≤
P
(
E
)
≤
1
for any event
E
;
2.
P
(
E does not occur
)
=
1
−
P
(
E
)
.
3. If
E
1
and
E
2
are mutually exclusive, then
P
(
E
1
or
E
2
)
=
P
(
E
1
)
+
P
(
E
2
)
.
3.1.2
Random Variable (Readings in Chapter 5)
Suppose there a random “mechanism” generates a lot of data, the
distribution
is how these
data look like if we plot them out by their values.
Here are some examples of distributions. Before we ±ip a coin, we know the possible out
comes are
{
heads, tails
}
.B
e
c
a
u
s
e
X
denotes a random event (before the coin is ±ipped), we
call
X
a
random variable
.
Defnition.
A
random variable
is a variable that takes on di
↵
erent values based on chance.
In the case of the coin, we know
X
has two outcomes,
x
1
=
“heads” and
x
2
=
“tails”. If the
coin is fair, we know that it will be heads half the time and be tails half the time. In other
words, if we denote the outcome of the coin ±ip by
X
,weknowthat
X
=
±
²
²
²
³
²
²
²
´
x
1
with probability
1
2
x
2
with probability
1
2
.
And the above “distribution” of the probabilities over outcomes is the
distribution
of the
random variable
X
.
The likelihood that the random event will be a particular outcome (“heads” and “tails”) is
called the
density
of the distribution.
In reality, there are a lot of uncertain events. Most of them have very complicated outcomes,
much more so than “heads” and “tails”. Therefore it is very useful to understand random
27
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Applied Statistical Methods: Lecture Notes (Leshui He)
2014
distributions with “continuous”, rather than “discrete”, outcomes. We will learn about the
most important continuous distribution: Normal distribution.
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 Fall '13
 Alexandru
 Normal Distribution, Probability

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