f(k,n)=f(k-1,n-1)*(1-n/N)+f(k,n-1)
f(1,n)=(1/N)^(n-1)
f(k,1)=0, if k<>1
E=sum(n*(f(7,n)-f(7,n-1)))
D=sum((n-E)^2*(f(7,n)-f(7,n-1)))
数值解法应该比较容易

用蒙特卡洛模拟可以吧？？

一般面试的时候就给你纸和笔，在面试官眼皮底下做。

let x_i be the first time to collect i games, which is a random variable.

then
E[x_7] = E[E[x_7 - x_6| x_6] + x_6 ] = 7 + E[x_6] = ... = \sum_{i = 1}^7 7/i

V[x_7] = V[x_7-x_6 + x_6 - x_5 + ... + x_2 - x_1] = V[x_7-x_6] + V[x_6-x_5]
+ ... + V[x_2-x_1] = 8/1 + 9/2 + ... + 13/6