Download PDF

multinomial distribution∗
CWoo†
2013-03-21 17:47:20
Let X = (X1 , . . . , Xn ) be a random vector such that
1. Xi ≥ 0 and Xi ∈ Z
2. X1 + · · · + Xn = N , where N is a fixed integer
Then X has a multinomial distribution if there exists a parameter vector π =
(π1 , . . . , πn ) such that
1. πi ≥ 0 and πi ∈ R
2. π1 + · · · + πn = 1
3. X has a discrete probability distribution function fX (x) in the form:
fX (x) =
n
Y
N!
π xi
x1 ! · · · xn ! i=1 i
Remarks
• E[X] = N π
• Var[X] = (vij ), where
(
vij =
N πi (1 − πi ) if i = j;
−N πi πj
if i =
6 j.
• When n = 2, the multinomial distribution is the same as the binomial
distribution
∗ hMultinomialDistributioni created: h2013-03-21i by: hCWooi version: h36113i Privacy
setting: h1i hDefinitioni h60E05i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
1
• If X1 , . . . , Xn are mutually independent Poisson random variables with
parameters λ1 , . . . , λn respectively, then the conditional joint distribution
of X1 , . . . , Xn given
P that X1 +· · ·+Xn = N is multinomial with parameters
λi /λ, where λ =
λi .
Sketch of proof. Each Xi is distributed as:
fXi (xi ) =
e−λi λxi i
xi !
The mutual independence of the Xi ’s shows that the joint probability
distribution of the Xi ’s is given by
fX (x) =
n
Y
e−λi λxi
i
i=1
xi !
= e−λ
n
Y
λxi
i
i=1
xi !
,
where X = (X1 , . . . , Xn ), x = (x1 , . . . , xn ) and λ = λ1 + · · · + λn . Next,
let X = X1 + · · · + Xn . Then X is Poisson distributed with parameter λ
(which can be shown by using induction and the mutual independence of
the Xi ’s):
e−λ λx
fX (x) =
.
x!
The conditional probability distribution of X given that X = N is thus
given by:
fX (x | X = N ) =
where
P
n
n
Y
Y
fX (x)
λi
λxi i
e−λ λN
N!
( )xi ,
= (e−λ
)/(
)=
fX (N )
x!
N!
x1 ! · · · xn ! i=1 λ
i=1 i
xi = N and that
P
λi /λ = 1.
2