第2講確率分布

有限母集団

$\theta_1, \theta_2, \ldots, \theta_N$ を大きさ $N$ の有限母集団とする．

P (X_{i} = θ_{k}) = \frac{_{N - 1} P_{n - 1}}{_{N} P_{n}} = \frac{1}{N}

$\begin{equation} P(X_i = \theta_k) = \frac{{}_{N-1}P_{n-1}}{{}_{N}P_{n}} = \frac{1}{N} \end{equation}$

P (X_{i} = θ_{k}, X_{j} = θ_{l}) = \frac{_{N - 2} P_{n - 2}}{_{N} P_{n}} = \frac{1}{N (N - 1)}

$\begin{equation} P(X_i = \theta_k, X_j = \theta_l) = \frac{{}_{N-2}P_{n-2}}{{}_{N}P_{n}} = \frac{1}{N(N-1)} \end{equation}$

μ = \frac{1}{N} \sum_{i = 1}^{N} θ_{i}

$\begin{equation} \mu = \frac{1}{N}\sum_{i=1}^{N}\theta_i \end{equation}$

σ^{2} = \frac{1}{N} \sum_{i = 1}^{N} (θ_{i} - μ)^{2}

$\begin{equation} \sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(\theta_i - \mu)^2 \end{equation}$