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多項分布 (Multinomial Distribution)
(1)
\begin{align} \frac{n!}{x_1!...x_k!} p_1^{x_1} p_2^{x_2}...p_k^{x_k} \end{align}
- 意義：n 次試驗中各種情況分別出現 x1, x2, …, xk 次的機率
- 範圍：x1, x2, …, xk=0,1,2,…,n ; 0<p[i]<1
- R 函數：multinom(size, prob) ; n:size:樣本數, p:prob:各種情況的機率
R 函數範例
rmultinom(10, size = 12, prob=c(0.1,0.2,0.8))
pr <- c(1,3,6,10) # normalization not necessary for generation
rmultinom(10, 20, prob = pr)
## all possible outcomes of Multinom(N = 3, K = 3)
X <- t(as.matrix(expand.grid(0:3, 0:3))); X <- X[, colSums(X) <= 3]
X <- rbind(X, 3:3 - colSums(X)); dimnames(X) <- list(letters[1:3], NULL)
X
round(apply(X, 2, function(x) dmultinom(x, prob = c(1,2,5))), 3)
執行結果：
> rmultinom(10, size = 12, prob=c(0.1,0.2,0.8))
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] 1 1 1 0 2 0 1 1 0 2
[2,] 1 2 3 0 3 0 2 1 1 2
[3,] 10 9 8 12 7 12 9 10 11 8
>
> pr <- c(1,3,6,10) # normalization not necessary for generation
> rmultinom(10, 20, prob = pr)
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] 1 1 2 2 1 1 2 1 1 2
[2,] 2 2 2 2 6 7 3 5 4 4
[3,] 9 4 8 4 8 8 4 7 3 6
[4,] 8 13 8 12 5 4 11 7 12 8
>
> ## all possible outcomes of Multinom(N = 3, K = 3)
> X <- t(as.matrix(expand.grid(0:3, 0:3))); X <- X[, colSums(X) <= 3]
> X <- rbind(X, 3:3 - colSums(X)); dimnames(X) <- list(letters[1:3], NULL)
> X
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
a 0 1 2 3 0 1 2 0 1 0
b 0 0 0 0 1 1 1 2 2 3
c 3 2 1 0 2 1 0 1 0 0
> round(apply(X, 2, function(x) dmultinom(x, prob = c(1,2,5))), 3)
[1] 0.244 0.146 0.029 0.002 0.293 0.117 0.012 0.117 0.023 0.016
>
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