二项式分布与伯努利分布

Bernoulli Experiment, Bernoulli Distribution, 0-1 Distribution

最常见的伯努利试验是抛一次硬币. 伯努利试验的结果服从伯努利分布: 随机变量只可能取0, 1两个值, 所以也称0-1分布.

\[p(X = x) = \begin{cases} \theta, x = 1\\ 1 -\theta, x = 0 \end{cases} \]

\[E(X) = \theta \]

\[D(X) = \theta (1 - \theta)^2 + (1 - \theta)\theta^2 = \theta(1 - \theta) \]

二项式分布, Binomial Distribution

\(n\)次独立的伯努利试验的结果服从二项式分布.

\[E(Y) = E(X_1 + X_2 + \dots + X_n) = nE(X) = n\theta \]

\[D(Y) = D(X_1 + X_2 + \dots + X_n) = nD(X) = n\theta(1 - \theta) \]