设打开门时已被用过的钥匙数为X,则X的分布列如下
X－P
1－1/n
2－1/n
3－1/n
.
.
n－1/n
EX
=1×1/n+2×1/n+3×1/n+...+n×1//n
=[(1+n)×n÷2]×1/n
=(1+n)/2
DX
=EX²-(EX)²
=(1²×1/n+2²×1/n+...+n²×1/n)-((1+n)/2)²
=[n(n+1)(2n+1)/6]/n-(1+n)²/4
=[(n+1)(2n+1)/6]-(n+1)²/4
=(n+1)(n-1)/12
=(n²-1)/12