Multinomial distribution

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CF

( ∑ j = 1 k p j e i t j ) n {\displaystyle \left(\sum _{j=1}^{k}p_{j}e^{it_{j}}\right)^{n}} where i 2 = − 1 {\displaystyle i^{2}=-1}

Entropy

− log ⁡ ( n ! ) − n ∑ i = 1 k p i log ⁡ ( p i ) + ∑ i = 1 k ∑ x i = 0 n ( n x i ) p i x i ( 1 − p i ) n − x i log ⁡ ( x i ! ) {\displaystyle {\begin{aligned}&-\log(n!)-n\sum _{i…

Mean

E ⁡ ( X i ) = n p i {\displaystyle \operatorname {E} (X_{i})=np_{i}}

MGF

( ∑ i = 1 k p i e t i ) n {\displaystyle \left(\sum _{i=1}^{k}p_{i}e^{t_{i}}\right)^{n}}

Parameters

n ∈ { 0 , 1 , 2 , … } {\displaystyle n\in \{0,1,2,\ldots \}} number of trials k > 0 {\displaystyle k>0} number of mutually exclusive events (integer) p 1 , … , p k {\displaystyl…

PGF

( ∑ i = 1 k p i z i ) n {\displaystyle \left(\sum _{i=1}^{k}p_{i}z_{i}\right)^{n}} for ( z 1 , … , z k ) ∈ C k {\displaystyle (z_{1},\ldots ,z_{k})\in \mathbb {C} ^{k}}