• Source: List of convolutions of probability distributions

In probability theory, the probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density functions respectively. Many well known distributions have simple convolutions. The following is a list of these convolutions. Each statement is of the form





∑

i
=
1


n



X

i


∼
Y


{\displaystyle \sum _{i=1}^{n}X_{i}\sim Y}


where




X

1


,

X

2


,
…
,

X

n




{\displaystyle X_{1},X_{2},\dots ,X_{n}}

are independent random variables, and



Y


{\displaystyle Y}

is the distribution that results from the convolution of




X

1


,

X

2


,
…
,

X

n




{\displaystyle X_{1},X_{2},\dots ,X_{n}}

. In place of




X

i




{\displaystyle X_{i}}

and



Y


{\displaystyle Y}

the names of the corresponding distributions and their parameters have been indicated.




Discrete distributions







∑

i
=
1


n



B
e
r
n
o
u
l
l
i

(
p
)
∼

B
i
n
o
m
i
a
l

(
n
,
p
)

0
<
p
<
1

n
=
1
,
2
,
…


{\displaystyle \sum _{i=1}^{n}\mathrm {Bernoulli} (p)\sim \mathrm {Binomial} (n,p)\qquad 0





∑

i
=
1


n



B
i
n
o
m
i
a
l

(

n

i


,
p
)
∼

B
i
n
o
m
i
a
l


(


∑

i
=
1


n



n

i


,
p

)


0
<
p
<
1


n

i


=
1
,
2
,
…


{\displaystyle \sum _{i=1}^{n}\mathrm {Binomial} (n_{i},p)\sim \mathrm {Binomial} \left(\sum _{i=1}^{n}n_{i},p\right)\qquad 0





∑

i
=
1


n



N
e
g
a
t
i
v
e
B
i
n
o
m
i
a
l

(

n

i


,
p
)
∼

N
e
g
a
t
i
v
e
B
i
n
o
m
i
a
l


(


∑

i
=
1


n



n

i


,
p

)


0
<
p
<
1


n

i


=
1
,
2
,
…


{\displaystyle \sum _{i=1}^{n}\mathrm {NegativeBinomial} (n_{i},p)\sim \mathrm {NegativeBinomial} \left(\sum _{i=1}^{n}n_{i},p\right)\qquad 0





∑

i
=
1


n



G
e
o
m
e
t
r
i
c

(
p
)
∼

N
e
g
a
t
i
v
e
B
i
n
o
m
i
a
l

(
n
,
p
)

0
<
p
<
1

n
=
1
,
2
,
…


{\displaystyle \sum _{i=1}^{n}\mathrm {Geometric} (p)\sim \mathrm {NegativeBinomial} (n,p)\qquad 0





∑

i
=
1


n



P
o
i
s
s
o
n

(

λ

i


)
∼

P
o
i
s
s
o
n


(


∑

i
=
1


n



λ

i



)



λ

i


>
0


{\displaystyle \sum _{i=1}^{n}\mathrm {Poisson} (\lambda _{i})\sim \mathrm {Poisson} \left(\sum _{i=1}^{n}\lambda _{i}\right)\qquad \lambda _{i}>0}





Continuous distributions







∑

i
=
1


n


Stable
⁡

(

α
,

β

i


,

c

i


,

μ

i



)

=
Stable
⁡

(

α
,




∑

i
=
1


n



β

i



c

i


α





∑

i
=
1


n



c

i


α





,


(


∑

i
=
1


n



c

i


α



)


1

/

α


,

∑

i
=
1


n



μ

i



)



{\displaystyle \sum _{i=1}^{n}\operatorname {Stable} \left(\alpha ,\beta _{i},c_{i},\mu _{i}\right)=\operatorname {Stable} \left(\alpha ,{\frac {\sum _{i=1}^{n}\beta _{i}c_{i}^{\alpha }}{\sum _{i=1}^{n}c_{i}^{\alpha }}},\left(\sum _{i=1}^{n}c_{i}^{\alpha }\right)^{1/\alpha },\sum _{i=1}^{n}\mu _{i}\right)}






0
<

α

i


≤
2

−
1
≤

β

i


≤
1


c

i


>
0

∞
<

μ

i


<
∞


{\displaystyle \qquad 0<\alpha _{i}\leq 2\quad -1\leq \beta _{i}\leq 1\quad c_{i}>0\quad \infty <\mu _{i}<\infty }


The following three statements are special cases of the above statement:





∑

i
=
1


n


Normal
⁡
(

μ

i


,

σ

i


2


)
∼
Normal
⁡

(


∑

i
=
1


n



μ

i


,

∑

i
=
1


n



σ

i


2



)


−
∞
<

μ

i


<
∞


σ

i


2


>
0

(
α
=
2
,

β

i


=
0
)


{\displaystyle \sum _{i=1}^{n}\operatorname {Normal} (\mu _{i},\sigma _{i}^{2})\sim \operatorname {Normal} \left(\sum _{i=1}^{n}\mu _{i},\sum _{i=1}^{n}\sigma _{i}^{2}\right)\qquad -\infty <\mu _{i}<\infty \quad \sigma _{i}^{2}>0\quad (\alpha =2,\beta _{i}=0)}






∑

i
=
1


n


Cauchy
⁡
(

a

i


,

γ

i


)
∼
Cauchy
⁡

(


∑

i
=
1


n



a

i


,

∑

i
=
1


n



γ

i



)


−
∞
<

a

i


<
∞


γ

i


>
0

(
α
=
1
,

β

i


=
0
)


{\displaystyle \sum _{i=1}^{n}\operatorname {Cauchy} (a_{i},\gamma _{i})\sim \operatorname {Cauchy} \left(\sum _{i=1}^{n}a_{i},\sum _{i=1}^{n}\gamma _{i}\right)\qquad -\infty 0\quad (\alpha =1,\beta _{i}=0)}






∑

i
=
1


n


Levy
⁡
(

μ

i


,

c

i


)
∼
Levy
⁡

(


∑

i
=
1


n



μ

i


,


(


∑

i
=
1


n





c

i





)


2



)


−
∞
<

μ

i


<
∞


c

i


>
0

(
α
=
1

/

2
,

β

i


=
1
)


{\displaystyle \sum _{i=1}^{n}\operatorname {Levy} (\mu _{i},c_{i})\sim \operatorname {Levy} \left(\sum _{i=1}^{n}\mu _{i},\left(\sum _{i=1}^{n}{\sqrt {c_{i}}}\right)^{2}\right)\qquad -\infty <\mu _{i}<\infty \quad c_{i}>0\quad (\alpha =1/2,\beta _{i}=1)}






∑

i
=
1


n


Gamma
⁡
(

α

i


,
β
)
∼
Gamma
⁡

(


∑

i
=
1


n



α

i


,
β

)



α

i


>
0

β
>
0


{\displaystyle \sum _{i=1}^{n}\operatorname {Gamma} (\alpha _{i},\beta )\sim \operatorname {Gamma} \left(\sum _{i=1}^{n}\alpha _{i},\beta \right)\qquad \alpha _{i}>0\quad \beta >0}






∑

i
=
1


n


Voigt
⁡
(

μ

i


,

γ

i


,

σ

i


)
∼
Voigt
⁡

(


∑

i
=
1


n



μ

i


,

∑

i
=
1


n



γ

i


,



∑

i
=
1


n



σ

i


2





)


−
∞
<

μ

i


<
∞


γ

i


>
0


σ

i


>
0


{\displaystyle \sum _{i=1}^{n}\operatorname {Voigt} (\mu _{i},\gamma _{i},\sigma _{i})\sim \operatorname {Voigt} \left(\sum _{i=1}^{n}\mu _{i},\sum _{i=1}^{n}\gamma _{i},{\sqrt {\sum _{i=1}^{n}\sigma _{i}^{2}}}\right)\qquad -\infty <\mu _{i}<\infty \quad \gamma _{i}>0\quad \sigma _{i}>0}






∑

i
=
1


n


VarianceGamma
⁡
(

μ

i


,
α
,
β
,

λ

i


)
∼
VarianceGamma
⁡

(


∑

i
=
1


n



μ

i


,
α
,
β
,

∑

i
=
1


n



λ

i



)


−
∞
<

μ

i


<
∞


λ

i


>
0




α

2


−

β

2




>
0


{\displaystyle \sum _{i=1}^{n}\operatorname {VarianceGamma} (\mu _{i},\alpha ,\beta ,\lambda _{i})\sim \operatorname {VarianceGamma} \left(\sum _{i=1}^{n}\mu _{i},\alpha ,\beta ,\sum _{i=1}^{n}\lambda _{i}\right)\qquad -\infty <\mu _{i}<\infty \quad \lambda _{i}>0\quad {\sqrt {\alpha ^{2}-\beta ^{2}}}>0}






∑

i
=
1


n


Exponential
⁡
(
θ
)
∼
Erlang
⁡
(
n
,
θ
)

θ
>
0

n
=
1
,
2
,
…


{\displaystyle \sum _{i=1}^{n}\operatorname {Exponential} (\theta )\sim \operatorname {Erlang} (n,\theta )\qquad \theta >0\quad n=1,2,\dots }






∑

i
=
1


n


Exponential
⁡
(

λ

i


)
∼
Hypoexponential
⁡
(

λ

1


,
…
,

λ

n


)


λ

i


>
0


{\displaystyle \sum _{i=1}^{n}\operatorname {Exponential} (\lambda _{i})\sim \operatorname {Hypoexponential} (\lambda _{1},\dots ,\lambda _{n})\qquad \lambda _{i}>0}






∑

i
=
1


n



χ

2


(

r

i


)
∼

χ

2



(


∑

i
=
1


n



r

i



)



r

i


=
1
,
2
,
…


{\displaystyle \sum _{i=1}^{n}\chi ^{2}(r_{i})\sim \chi ^{2}\left(\sum _{i=1}^{n}r_{i}\right)\qquad r_{i}=1,2,\dots }






∑

i
=
1


r



N

2


(
0
,
1
)
∼

χ

r


2



r
=
1
,
2
,
…


{\displaystyle \sum _{i=1}^{r}N^{2}(0,1)\sim \chi _{r}^{2}\qquad r=1,2,\dots }






∑

i
=
1


n


(

X

i


−



X
¯




)

2


∼

σ

2



χ

n
−
1


2


,



{\displaystyle \sum _{i=1}^{n}(X_{i}-{\bar {X}})^{2}\sim \sigma ^{2}\chi _{n-1}^{2},\quad }

where




X

1


,
…
,

X

n




{\displaystyle X_{1},\dots ,X_{n}}

is a random sample from



N
(
μ
,

σ

2


)


{\displaystyle N(\mu ,\sigma ^{2})}

and






X
¯



=


1
n



∑

i
=
1


n



X

i


.


{\displaystyle {\bar {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}.}


Mixed distributions:




Normal
⁡
(
μ
,

σ

2


)
+
Cauchy
⁡
(

x

0


,
γ
)
∼
Voigt
⁡
(
μ
+

x

0


,
σ
,
γ
)

−
∞
<
μ
<
∞

−
∞
<

x

0


<
∞

γ
>
0

σ
>
0


{\displaystyle \operatorname {Normal} (\mu ,\sigma ^{2})+\operatorname {Cauchy} (x_{0},\gamma )\sim \operatorname {Voigt} (\mu +x_{0},\sigma ,\gamma )\qquad -\infty <\mu <\infty \quad -\infty 0\quad \sigma >0}





See also


Algebra of random variables
Relationships among probability distributions
Infinite divisibility (probability)
Bernoulli distribution
Binomial distribution
Cauchy distribution
Erlang distribution
Exponential distribution
Gamma distribution
Geometric distribution
Hypoexponential distribution
Lévy distribution
Poisson distribution
Stable distribution
Mixture distribution
Sum of normally distributed random variables




References






Sources


Hogg, Robert V.; McKean, Joseph W.; Craig, Allen T. (2004). Introduction to mathematical statistics (6th ed.). Upper Saddle River, New Jersey: Prentice Hall. p. 692. ISBN 978-0-13-008507-8. MR 0467974.

Kata Kunci Pencarian:

• List of convolutions of probability distributions
• Convolution of probability distributions
• List of probability distributions
• Relationships among probability distributions
• List of statistics articles
• Convolution
• Sum of normally distributed random variables
• Distribution of the product of two random variables
• Normal distribution
• Mixture distribution