====== 베르누이 분포 (Bernoulli Distribution) ======
===== 정의 =====
===== 표기 =====
 $$ X \sim b(1 , p)$$

  * $$ p \in [ \ 0 \ , \ 1 \ ] $$
===== 받침 =====
 $$ x \in \{ \ 0 \ , \ 1 \ \} $$
===== 확률질량함수 =====
 $$p(x)=p^{x}(1-p)^{1-x}$$

<plot>
 set title "Bernoulli Distribution PMF"
 set size 1.0
 set xtics (0,1)
 set yrange [0:1]
 set xrange [-0.5:1.5]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "p(x)"

 f(x,p) = (p**(int(x)))*((1-p)**(1-(int(x))))

 plot f(x+0.5,0.4) title "b(1,0.4)" with steps
</plot>
===== 누적분포함수 =====
 $$ F(x) = (1 - p)^{1 - x} $$

<plot>
 set title "Bernoulli Distribution CDF"
 set size 1.0
 set xtics (0,1)
 set yrange [0:1.1]
 set xrange [-0.5:1.5]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "F(x)"
 set key left

 f(x,p) = ((1-p)**(1-(int(x))))

 plot f(x+0.5,0.4) title "b(1,0.4)" with steps
</plot>
===== 기대값 =====
 $$E(X)=p$$
===== 분산 =====
 $$Var(X)=p(1-p)$$
===== 왜도 =====
 $$ \gamma_{ \ 1} = \frac{1 - 2p}{\sqrt{p(1 - p)}} = \frac{q - p}{\sqrt{pq}} $$
===== 첨도 =====
 $$ \gamma_{ \ 2} = \frac{6p^{2} - 6p + 1}{p(1 - p)} = \frac{1 - 6pq}{pq} $$
===== 특성함수 =====
 $$ \phi \ (t) = 1 + p(e^{it} - 1) $$
===== 적률생성함수 =====
  * $$M(t)=pe^{t}+(1-p)$$
    * $$ M'(t) = pe^{t} $$
    * $$ M''(t) = pe^{t} $$
    * $$ M^{(n)}(t) = pe^{t} $$
===== 원적률 =====
  * $$ \mu'_{1} = p $$
  * $$ \mu'_{2} = p $$
  * $$ \mu'_{n} = p $$
===== 중심적률 =====
  * $$ \mu_{2} = p(1 - p) $$
  *  $$ \mu_{3} = p(1 - p)(1 - 2p) $$
  *  $$ \mu_{4} = p(1 - p)(3p^{2} - 3p + 1) $$

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  * [[분포]] 
  * [[이항분포]]