2項係数から確率分布を作る - keisuke araki's project

2項係数から確率分布を作る

2項分布から作った確率分布$ P_k = \frac{{}_NC_k}{2^N} = w_n h_k ここで$ w_N ,$ h_k は分布関数の短冊の幅、高さ

$ \sum_{k=0}^{N} P(k) = \frac{\displaystyle\lim_{x\to1}(1+x)^N}{2^N} = \frac{\displaystyle\sum_{k=0}^{N}{}_NC_kx^k}{2^N} = 1

1次モーメント

$ \sum_{k=0}^{N} k P(k) = \frac{\displaystyle\lim_{x\to1}N(1+x)^{N-1}}{2^N} = \frac{\displaystyle\sum_{k=0}^{N}k\,{}_NC_kx^{k-1}}{2^N} = \frac{N}{2} $ \Longrightarrow $ 0 = \sum_{k=0}^{N} k P(k) - \frac{N}{2} = \sum_{k=0}^{N} (k- \frac{N}{2}) P(k)

2次モーメント

$ \sum_{k=0}^{N} k(k-1) P(k) = \frac{\displaystyle\lim_{x\to1}N(N-1)(1+x)^{N-2}}{2^N} = \frac{\displaystyle\sum_{k=0}^{N}k(k-1)\,{}_NC_kx^{k-2}}{2^N} = \frac{N(N-1)}{2^2}

より

$ \sum_{k=0}^{N} k^2 P(k) = \sum_{k=0}^{N} k(k-1) P(k) + \sum_{k=0}^{N} k P(k) = \frac{N^2+N}{4}

$ \sum_{k=0}^{N} (k-\frac{N}{2})^2 P(k) = \sum_{k=0}^{N} \left\{k(k-\frac N2) - \frac N2(k-\frac N2)\right\} P(k) = \frac{N^2+N}{4} - \frac N2 \times \frac N2 = \frac N4

$ \Longrightarrow $ 1 = \sum_{k=0}^{N} \frac 4N (k-\frac{N}{2})^2 P(k) = \sum_{k=0}^{N} \left(\frac{k-N/2}{2\sqrt{N}}\right)^2 P(k) $ \Longrightarrow 幅$ w_N = \frac{1}{2\sqrt N} $ \Longrightarrow 高さ$ h_k = \frac{2\sqrt N{}_NC_k}{2^N}