normal random variable

normal random variable

For any real numbers

μ

and

σ

>

0

, the

Gaussian probability distribution

function

with mean

μ

and

variance

σ

2

is defined by

f

⁢

(

x

)

=

1

2

⁢

π

⁢

σ

2

⁢

exp

⁡

(

-

1

2

⁢

(

x

-

μ

σ

)

2

)

.

When

μ

=

0

and

σ

=

1

,
it is usually called

standard normal distribution

.

A

random variable

X

having

distribution

density

f

is said to be a

normally distributed random variable

, denoted by

X

∼

N

⁢

(

μ

,

σ

2

)

.
It has

expected value

μ

,
and variance

σ

2

.

Cumulative distribution function

The

cumulative distribution function

of a standard normal variable,
often denoted by

Φ

⁢

(

z

)

=

1

2

⁢

π

⁢

∫

-

∞

z

e

-

x

2

/

2

⁢

𝑑

x

,

cannot be calculated in closed form in terms of the elementary functions,
but its values are tabulated in most

statistics

books and here (

http://planetmath.org/TableOfProbabilitiesOfStandardNormalDistribution

),
and can be computed using most

computer

statistical packages and spreadsheets.

Uses of the Gaussian distribution

The

normal distribution

is probably the most frequently used distribution.
Its graph looks like a bell-shaped function, which is why it is often
called

bell distribution

.

The normal distribution is important in probability theory and statistics.
Empircally, many observed distributions,
such as of people’s heights, test scores,
experimental errors, are found to be more or less to be Gaussian.
And theoretically, the normal distribution arises as a limiting
distribution of averages of large numbers of samples, justified
by the

central limit theorem

.

Figure 1:

Graph of densities of the normal
distribution for various values of the

standard deviation

σ

Properties

Mean

μ

Variance

σ

2

Skewness

0

Kurtosis

3

Moment-generating function

M

X

⁢

(

t

)

=

exp

⁡

(

μ

⁢

t

+

(

σ

⁢

t

)

2

/

2

)

Characteristic function

ϕ

X

⁢

(

t

)

=

exp

⁡

(

μ

⁢

i

⁢

t

-

(

σ

⁢

t

)

2

/

2

)

•

If

Z

is a standard normal random variable,
then

X

=

σ

⁢

Z

+

μ

is distributed as

N

⁢

(

μ

,

σ

2

)

,
and

conversely

.

•

The sum of any finite number of

independent

normal

variables

is itself
a normal random variable.

Relations to other distributions

1.

The standard normal distribution can be considered as a Student-t distribution
with

infinite

degrees of freedom

.

2.

The square of the standard normal random variable is the

chi-squared random variable

of degree 1. Therefore, the sum of squares of

n

independent standard normal random variables is the chi-squared random variable of degree

n

.

Title

normal random variable

Canonical

name

NormalRandomVariable

Date of creation

2013-03-22 11:54:20

Last modified on

2013-03-22 11:54:20

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

22

Author

Koro (127)

Entry type

Definition

Classification

msc 62E15

Classification

msc 60E05

Classification

msc 05C50

Classification

msc 34K05

Synonym

normal distribution

Synonym

standard normal distribution

Synonym

bell distribution

Synonym

bell curve

Synonym

Gaussian

Related topic

AreaUnderGaussianCurve

Related topic

JointNormalDistribution

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