μ΄νλΆν¬λ 'μ±κ³΅'κ³Ό 'μ€ν¨'μ λ κ°μ§ μμ±λ§μ κ°λ λͺ¨μ§λ¨μ λμμΌλ‘ νμ§λ§, λ€νλΆν¬λ μΈ κ°μ§ μ΄μμ μμ±μ κ°λ λͺ¨μ§λ¨μμ λ°μνλ€.
kκ°μ μμ±μ΄ μ‘΄μ¬νκ³ , κ° μμ±μ λΉμ¨μ΄ p1β,p2β,...,pkβ μΈ λ¬΄ν λͺ¨μ§λ¨μμ n κ°μ νλ³Έμ μΆμΆν κ²½μ°, κ° μμ±μ κ°―μλ₯Ό X1β,X2β,...,Xkβ λΌ νλ©΄, λ€μκ³Ό κ°μ΄ λ€νλΆν¬λ₯Ό λ°λ₯Έλ€.
MN(n,p1β,p2β,...,pkβ)
λ€νλΆν¬(Multinomial Distribution)
P(X1β=x1β,X2β=x2β,...Xkβ=xkβ)=(x1β)!(x2β)!...(xkβ)!n!βp1x1ββp2x2ββ...pkxkββ
Where, Ξ£i=1kβpiβ=1, Ξ£i=1kβxiβ=n, 0β€xiββ€n.
(x1β)!(x2β)!...(xkβ)!n!βλ n κ°μ νλ³Έ μ€ κ°κ°μ μμ±μ΄ x1β,x2β,...,xkβ κ°μ© λμ€λ κ²½μ°μ μμ΄λ©°, p1x1ββp2x2ββ...pkxkββ λ νΉμ μ‘°ν©μ νλ₯ μμ μ μ μλ€.
λ€νλΆν¬μ νΉμ± μ€ νλλ Xiβ μ μ£Όλ³νλ₯ (marginal probability) λΆν¬κ° μ΄νλΆν¬κ° λλ€λ μ¬μ€μ΄λ€. νΉμ ν i λ²μ§Έ μμ±λ§μ 'μ±κ³΅'μΌλ‘ μ νλ©΄ μ 체 λͺ¨μ§λ¨μ 'μ±κ³΅'κ³Ό 'μ€ν¨'μ λ κ°μ§ μμ±μΌλ‘ λλ μ μκΈ° λλ¬Έμ΄λ€. λ°λΌμ Xiβμ μ£Όλ³νλ₯ λΆν¬λ λ€μκ³Ό κ°λ€.
P(Xiβ=xiβ)=(xiβnβ)pixiββ(1βp)nβxiβ, xiβ=0,1,...,n.
Expected Value and Variance of Multinomial Distribution
E(Xiβ)=npiβ, Var(Xiβ)=npiβ(1βpiβ).
EXAMPLE 13. μΈ κ°μ§ μμ±μ κ°λ μλμ μΈ λͺ¨μ§λ¨μμ κ°κ° 5κ°μ© νλ³Έμ μ·¨νμμ λ, λ€ν νλ₯ λΆν¬ κ·Έλνλ₯Ό μμ±νκ³ , κΈ°λκ° λΆμ°μ ꡬνμ¬ λΉκ΅νμμ€.
(p1β,p2β,p3β)=(0.1,0.1,0.8)
(p1β,p2β,p3β)=(0.1,0.5,0.4)
(p1β,p2β,p3β)=(1/3,1/3,1/3)
[ Solution ]
E(X1β)=5Γ0.1=0.5, E(X2β)=5Γ0.1=0.5, E(X3β)=5Γ0.8=0.4
Var(X1β)=5Γ0.09=0.45, Var(X2β)=5Γ0.09=0.45,
Var(X3β)=5Γ0.16=0.8
E(X1β)=5Γ0.1=0.5, E(X2β)=5Γ0.5=2.5, E(X3β)=5Γ0.4=2
Var(X1β)=5Γ0.09=0.45, Var(X2β)=5Γ0.25=1.25,
Var(X3β)=5Γ0.24=1.2
E(X1β)=5/3, E(X2β)=5/3, E(X3β)=5/3
Var(X1β)=Var(X2β)=Var(X3β)=5Γ(1/3)Γ(2/3)=10/9β1.111
library(Rstat)
# pi, n, range of xi
p <- matrix(c(1, 1, 8, 1, 5, 4, 1, 1, 1), nrow=3, ncol=3, byrow=T)
# Packages : scatterplot3d
# install.packages("scatterplot3d")
library(scatterplot3d)
# multinorm.plot()
multinorm.plot(p, 5)
EXAMPLE 14. μ΄λ€ νλ‘μΈμ€μμ μμ°λλ μ νμ νμ§μ΄ A, B, C, D λ±κΈμΌλ‘ ꡬλΆλλ©°, κ° λ±κΈμ λΉμ¨μ 20%, 40%, 30%, 10%λ‘ μλ €μ Έ μλ€. μ΄ κ³΅μ μμ 20κ°μ μ νμ μνλ§νμμ λ, κ° λ±κΈ μ νμ μλ₯Ό X1β,X2β,X3β,X4βλΌ νμ.
νλ₯ λΆν¬ν¨μλ₯Ό ꡬνλΌ.
κΈ°λκ°κ³Ό λΆμ°μ ꡬνλΌ.
X1β=3,X2β=6,X3β=8 μ΄ λμ¬ νλ₯ μ ꡬνλΌ.
X1β=3,X2β=6 μ΄ λμ¬ νλ₯ μ ꡬνλΌ.
X1β=3 μ΄ λμ¬ νλ₯ μ ꡬνλΌ.
[ Solution ]
f(x1β,x2β,x3β,x4β)=x1β!x2β!x3β!x4β!20!β(0.2)x1β(0.4)x2β(0.3)x3β(0.1)x4β
E(X1β)=20Γ0.2=4, E(X2β)=8, E(X3β)=6, E(X4β)=2
Var(X1β)=20Γ0.2Γ0.8=3.2, Var(X2β)=4.8, Var(X3β)=4.2, Var(X4β)=1.8
X1β=3,X2β=6,X3β=8 μ΄λ©΄, X4β=3.
f(3,6,8,3)=3!Γ6!Γ8!Γ3!20!β(0.2)3(0.4)6(0.3)8(0.1)3β0.005
X1β=3,X2β=6 μ΄λ©΄, λλ¨Έμ§ μμ±μ 11.
f(3,6,11)=3!Γ6!Γ11!20!β(0.2)3(0.4)6(0.4)11β0.019
X1β=3μ΄λ©΄, λλ¨Έμ§ μμ±μ 17.
f(3,17)=3!Γ17!20!β(0.2)3(0.8)17β0.205
library(Rstat)
# pi, n, range of xi
p <- c(2, 4, 3, 1)
n <- 20
# 3. f(3, 6, 8, 3)
x <- c(3, 6, 8, 3)
dmultinom(x, size=n, prob=p)
# 4. f(3, 6, 11)
x <- c(3, 6, 11)
dmultinom(x, size=n, prob=c(2, 4, 4))
# 5. f(3, 17)
x <- c(3, 17)
dmultinom(x, size=n, prob=c(2,8))
> # 3. f(3, 6, 8, 3)
> x <- c(3, 6, 8, 3)
> dmultinom(x, size=n, prob=p)
## [1] 0.005004827
> # 4. f(3, 6, 11)
> x <- c(3, 6, 11)
> dmultinom(x, size=n, prob=c(2, 4, 4))
## [1] 0.01939077
> # 5. f(3, 17)
> x <- c(3, 17)
> dmultinom(x, size=n, prob=c(2,8))
## [1] 0.2053641