PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
Possible values:
Probability function:
Integer valued (x 1 , x 2 , . . . , x r ) s.t. x i ≥ 0 &
r∑
x i = n
i=1
(
)
n
P (x) =
π x 1
1 π x 2
2 . . . πr
xr
x 1 , x 2 , . . . , x r
for
x i ≥ 0,
r∑
x i = n.
i=1
Remarks
(
)
n
1. Note
x 1 , x 2 , . . . , x r
def
=
n!
x 1 !x 2 ! . . . x r !
is the multinomial coefficient.
2. Multinomial distribution is the generalisation of the binomial distribution to r
types of outcome.
3. Formulation differs from binomial and trinomial cases in that the redundant count
x r = n − (x 1 + x 2 + . . . x r−1 ) is included as an argument of P (x).
1.7.3 Marginal and conditional distributions
Consider a discrete random vector
so that X =
[ ]
X1
.
X 2
X = (X 1 , X 2 , . . . , X r ) T and let
X 1 = (X 1 , X 2 , . . . , X r1 ) T &
X 2 = (X r1 +1, X r1 +2, . . . , X r ) T ,
Definition. 1.7.2
If X has joint probability function P X (x) = P X (x 1 , x 2 ) then the marginal probability
function for X 1 is :
P X1 (x 1 ) = ∑ x 2
P X (x 1 , x 2 ).
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