クーポンコレクター問題
確率変数 ti の独立性を用いて、分散が以下のように計算できる。
Var  ( T ) = Var  ( t 1 ) + Var  ( t 2 ) + ⋯ + Var  ( t n ) = 1 − p 1 p 1 2 + 1 − p 2 p 2 2 + ⋯ + 1 − p n p n 2 < ( n 2 n 2 + n 2 ( n − 1 ) 2 + ⋯ + n 2 1 2 ) = n 2 ⋅ ( 1 1 2 + 1 2 2 + ⋯ + 1 n 2 ) < π 2 6 n 2 {\displaystyle {\begin{aligned}\operatorname {Var} (T)&=\operatorname {Var} (t_{1})+\operatorname {Var} (t_{2})+\cdots +\operatorname {Var} (t_{n})\\&={\frac {1-p_{1}}{p_{1}^{2}}}+{\frac {1-p_{2}}{p_{2}^{2}}}+\cdots +{\frac {1-p_{n}}{p_{n}^{2}}}\\&<\left({\frac {n^{2}}{n^{2}}}+{\frac {n^{2}}{(n-1)^{2}}}+\cdots +{\frac {n^{2}}{1^{2}}}\right)\\&=n^{2}\cdot \left({\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+\cdots +{\frac {1}{n^{2}}}\right)\\&<{\frac {\pi ^{2}}{6}}n^{2}\end{aligned}}}