====== 초기하분포 (Hypergeometric Distribution) ======
===== 정의 =====
===== 표기 =====
 $N$ : [[모집단]] 전체 [[원소]]의 개수 \\
 $M$ : [[모집단]]에서 특정 속성을 지닌 [[원소]]의 개수 \\
 $n$ : [[모집단]]에서 [[비복원추출]]로 뽑는 [[원소]]의 개수

  * $$ X \sim HG(N , M , n) $$
    * $$ N \in \{ \ 0 \ , \ 1 \ , \ \cdots \ \} $$
    * $$ M \in \{ \ 0 \ , \ 1 \ , \  \cdots \ , \ N \ \} $$
    * $$ n \in \{ \ 0 \ , \ 1 \ , \ \cdots \ , \ N \ \} $$
===== 받침 =====
 $$ x \in \{ \ 0 \ , \ 1 \ , \ \cdots \ , \ n \} $$
===== 확률질량함수 =====
 $$p(x)=\frac{\begin{pmatrix}M\\x\end{pmatrix} \begin{pmatrix}N-M\\n-x\end{pmatrix}}{\begin{pmatrix}N\\n\end{pmatrix}}$$

<plot>
 set title "Hypergeometric Distribution PMF (1)"
 set size 1.0
 set xrange [-0.5:5.5]
 set yrange [0:0.5]
 set format y "%.2f"
 set xlabel "x"
 set ylabel "p(x)"

 f(x,l,m,n) = (((m!)/((int(x)!)*((m-int(x))!)))*(((l-m)!)/(((n-int(x))!)*(((l-m)-(n-int(x)))!))))/((l!)/((n!)*((l-n)!)))

 plot f(x+0.5,20,8,3) with steps title "HG(20,8,1)", \
  f(x+0.5,20,8,5) with steps title "HG(20,8,3)", \
  f(x+0.5,20,8,7) with steps title "HG(20,8,5)"
</plot>
<plot>
 set title "Hypergeometric Distribution PMF (2)"
 set size 1.0
 set xrange [-0.5:10.5]
 set yrange [0:0.3]
 set format y "%.2f"
 set xlabel "x"
 set ylabel "p(x)"

 f(x,l,m,n) = (((m!)/((int(x)!)*((m-int(x))!)))*(((l-m)!)/(((n-int(x))!)*(((l-m)-(n-int(x)))!))))/((l!)/((n!)*((l-n)!)))

 plot f(x+0.5,100,50,10) with steps title "HG(100,50,10)"
</plot>
===== 누적분포함수 =====
===== 기대값 =====
 $$E(X)=\frac{nM}{N}$$
===== 분산 =====
 $$Var(X)=n \left( \frac{M}{N} \right) \left( \frac{N-M}{N} \right) \left( \frac{N-n}{N-1} \right) $$
===== 왜도 =====
===== 첨도 =====
===== 타 분포와의 관계 =====
  * [[초기하분포를 이항분포로 근사]]

----
  * [[분포]]
  * [[이산형 분포]]
  * [[비복원추출]]