sums of normal random variables need not be normal

sums of normal random variables need not be normal

A common misconception among students of probability theory is the belief that the sum of two normally distributed (

http://planetmath.org/NormalRandomVariable

)

random variables

is itself normally distributed. By constructing a counterexample, we show this to be false.

It is however well known that the sum of normally distributed variables

X

,

Y

will be normal under either of the following situations.

•

X

,

Y

are

independent

.

•

X

,

Y

are joint normal (

http://planetmath.org/JointNormalDistribution

).

The statement that the sum of two independent

normal random variables

is itself normal is a very useful and often used property. Furthermore, when working with normal variables which are not independent, it is common to suppose that they are in fact joint normal. This can lead to the belief that this property holds always.

Another common

fallacy

, which our example shows to be false, is that normal random variables are independent if and only if their

covariance

, defined by

Cov

⁡

(

X

,

Y

)

≡

𝔼

⁢

[

X

⁢

Y

]

-

𝔼

⁢

[

X

]

⁢

𝔼

⁢

[

Y

]

is zero. While it is certainly true that independent variables have zero covariance, the

converse

statement does not hold. Again, it is often supposed that they are joint normal, in which case a zero covariance will indeed imply independence.

We construct a pair of random variables

X

,

Y

satisfying the following.

1.

X

and

Y

each have the

standard normal distribution

.

2.

The covariance,

Cov

⁡

(

X

,

Y

)

, is zero.

3.

The sum

X

+

Y

is not normally distributed, and

X

and

Y

are not independent.

We start with a pair of independent random variables

X

,

ϵ

where

X

has the standard normal distribution and

ϵ

takes the values

1

,

-

1

, each with a

probability

of

1

/

2

.
Then set,

Y

=

{

ϵ

⁢

X

,

if

⁢

|

X

|

≤

1

,

-

ϵ

⁢

X

,

if

⁢

|

X

|

>

1

.

If

S

is any measurable subset of the real numbers, the

symmetry

of the

normal distribution

implies that

ℙ

(

X

∈

S

)

is equal to

ℙ

(

-

X

∈

S

)

. Then, by the independence of

X

and

ϵ

, the

distribution

of

Y

conditional

on

ϵ

=

1

is given by,

ℙ

(

Y

∈

S

∣

ϵ

=

1

)

=

ℙ

(

|

X

|

≤

1

,

X

∈

S

)

+

ℙ

(

|

X

|

>

1

,

-

X

∈

S

)

=

ℙ

(

|

X

|

≤

1

,

X

∈

S

)

+

ℙ

(

|

X

|

>

1

,

X

∈

S

)

=

ℙ

(

X

∈

S

)

.

It can similarly be shown that

ℙ

(

Y

∈

S

∣

ϵ

=

-

1

)

is equal to

ℙ

(

X

∈

S

)

.
So,

Y

has the same distribution as

X

and is normal with mean zero and

variance

one.

Using the fact that

ϵ

has zero mean and is independent of

X

, it is easily shown that the covariance of

X

and

Y

is zero.

Cov

⁡

(

X

,

Y

)

=

𝔼

⁢

[

X

⁢

Y

]

-

𝔼

⁢

[

X

]

⁢

𝔼

⁢

[

Y

]

=

𝔼

⁢

[

X

⁢

Y

]

=

𝔼

⁢

[

1

{

|

X

|

≤

1

}

⁢

ϵ

⁢

X

2

]

+

𝔼

⁢

[

-

1

{

|

X

|

>

1

}

⁢

ϵ

⁢

X

2

]

=

𝔼

⁢

[

ϵ

]

⁢

𝔼

⁢

[

1

{

|

X

|

≤

1

}

⁢

X

2

]

-

𝔼

⁢

[

ϵ

]

⁢

𝔼

⁢

[

1

{

|

X

|

>

1

}

⁢

X

2

]

=

0

.

As

X

and

Y

have zero covariance and each have variance equal to

1

, the sum

X

+

Y

will have variance equal to

2

. Also, the sum

satisfies

X

+

Y

=

{

2

⁢

ϵ

⁢

X

,

if

⁢

|

X

|

≤

1

,

0

,

if

⁢

|

X

|

>

1

.

In particular, this shows that

ℙ

(

|

X

+

Y

|

>

2

)

=

0

. However, normal random variables with nonzero variance always have a

positive

probability of being greater than any given real number. So,

X

+

Y

is not normally distributed.

This also shows that, despite having zero covariance,

X

and

Y

are not independent. If they were, then the fact that sums of independent normals are normal would imply that

X

+

Y

is normal, contradicting what we have just demonstrated.

Title

sums of normal random variables need not be normal

Canonical name

SumsOfNormalRandomVariablesNeedNotBeNormal

Date of creation

2013-03-22 18:43:44

Last modified on

2013-03-22 18:43:44

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Example

Classification

msc 62E15

Classification

msc 60E05

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