Multinomial Distribution

Multinomial Distribution - Definition

• $P(n_1,n_2,...,n_k|n)=\frac{n!}{(n_1!)(n_2!)...(n_k!)}{p}_1^{n_1}{p}_2^{n_2}...p_k^{n_k}$

Its parameters are:

• 𝑛 - the fixed total number of multinoulli trials
• 𝑝_𝑖 - the fixed probability of outcome 𝑖 for each multinoulli trial
  • 𝛴_{1≤𝑖≤𝑘}[𝑝_𝑖] = 1
• 𝑛_𝑖 - is the number of times outcome 𝑖 occurs in question
  • 𝛴_{1≤𝑖≤𝑘}[𝑛_𝑖] = 𝑛
• Each multinoulli trial has 𝑘 outcomes

Multinomial Distribution - Example

For the chess example:

• 𝑛 = 12 (12 games are played)
• 𝑛₁ = 7 (number won by Player A)
• 𝑛₂= 2 (number won by Player B)
• 𝑛₃ = 3 (the number drawn)
• 𝑝₁ = 0.40 (probability Player A wins)
• 𝑝₂ = 0.35 (probability Player B wins)
• 𝑝₃ = 0.25 (probability of a draw)

Thus:

• $P(7,2,3|12)=\frac{12!}{(7!)(2!)(3!)}0.40^{7}0.35^{2}0.25^3=0.0248$