Bernoulli Experiment, Bernoulli Distribution, 0-1 Distribution
最常見的伯努利試驗是拋一次硬幣. 伯努利試驗的結果服從伯努利分布: 隨機變量只可能取0, 1兩個值, 所以也稱0-1分布.
\[p(X = x) = \begin{cases} \theta, x = 1\\ 1 -\theta, x = 0 \end{cases} \]
\[E(X) = \theta \]
\[D(X) = \theta (1 - \theta)^2 + (1 - \theta)\theta^2 = \theta(1 - \theta) \]
二項式分布, Binomial Distribution
\(n\)次獨立的伯努利試驗的結果服從二項式分布.
\[E(Y) = E(X_1 + X_2 + \dots + X_n) = nE(X) = n\theta \]
\[D(Y) = D(X_1 + X_2 + \dots + X_n) = nD(X) = n\theta(1 - \theta) \]