Multinomial Distribution

Definition (Multinomial Distribution).

independent and identical experiments. Each experiments results in one of possible outcomes, with probability such that
is number of results in outcome
whenever

this is also written as the Multinomial

this is called the Multinomial

Remark (Multinomial as Binomial).

when Multinomial is same as Binomial

Remark.

Definition (Independence of Random Variable ).

Let take values be
let take values
The random variables and are independent if

Remark.

Equivalently are independent if for all

Theorem.

and are independent if and only if

Proof.

let

The other way

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Example

Exercise.

Given random number of balls of Poisson Poisson ()
Each ball is colored white with probability and Black with probability
let
Total number of balls
number of white balls
number of black balls

Solution:

Remark.

If number of balls fixed say then clearly is clearly not independent

Remark.

if Then are independent from^716f78 (1.1)

Exercise.

Let number of balls Geometric

Color each ball white with probability
Color each ball black with probability
Number of white balls
Number of black balls
show that is not independent

Exercise (Sum of Poisson Variables).

show that

Exercise (Sum of Binomials).

show that

have joint pmf

we saw earlier

Theorem.

If and are independent then

for some

Proof.

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