베르누이 분포 (Bernoulli Distribution)

$$ X \sim b(1 , p)$$

• $$ p \in [ \ 0 \ , \ 1 \ ] $$

$$ x \in \{ \ 0 \ , \ 1 \ \} $$

$$p(x)=p^{x}(1-p)^{1-x}$$

$$ F(x) = (1 - p)^{1 - x} $$

$$E(X)=p$$

$$Var(X)=p(1-p)$$

$$ \gamma_{ \ 1} = \frac{1 - 2p}{\sqrt{p(1 - p)}} = \frac{q - p}{\sqrt{pq}} $$

$$ \gamma_{ \ 2} = \frac{6p^{2} - 6p + 1}{p(1 - p)} = \frac{1 - 6pq}{pq} $$

$$ \phi \ (t) = 1 + p(e^{it} - 1) $$

• $$M(t)=pe^{t}+(1-p)$$
  • $$ M'(t) = pe^{t} $$
  • $$ M''(t) = pe^{t} $$
  • $$ M^{(n)}(t) = pe^{t} $$

• $$ \mu'_{1} = p $$
• $$ \mu'_{2} = p $$
• $$ \mu'_{n} = p $$

• $$ \mu_{2} = p(1 - p) $$
• $$ \mu_{3} = p(1 - p)(1 - 2p) $$
• $$ \mu_{4} = p(1 - p)(3p^{2} - 3p + 1) $$

• 분포
• 이항분포