第十題
求\(\displaystyle 1\cdot 3 C_1^{16}\left(\frac{3}{4}\right)\left(\frac{1}{4}\right)^{15}+2\cdot 4\cdot C_2^{16}\left(\frac{3}{4}\right)^{2}\left(\frac{1}{4}\right)^{14}+3\cdot 5\cdot C_3^{16}\left(\frac{3}{4}\right)^{3}\left(\frac{1}{4}\right)^{13}+\ldots+16\cdot 18\cdot C_{16}^{16}\left(\frac{1}{4}\right)^{16}\)之值 。
[解答]
考慮一個隨機變數X服從二項式分配，n=16，p=3/4
E(X)=12，Var(X)=3
E(X^2)=Var(X)+[E(X)]^2=3+144=147
原式=E(X^2)+2E(X)=147+24=171
