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