Let 
be the random variable indicating whether the elevator does not stop at floor 
, with
Let
be the random variable representing the number of floors at which the elevator does not stop. Then
We want to find
. By definition,
![\mathrm{Var}(Y)=\mathbb E[(Y-\mathbb E[Y])^2]=\mathbb E[Y^2]-\mathbb E[Y]^2](https://tex.z-dn.net/?f=%5Cmathrm%7BVar%7D%28Y%29%3D%5Cmathbb%20E%5B%28Y-%5Cmathbb%20E%5BY%5D%29%5E2%5D%3D%5Cmathbb%20E%5BY%5E2%5D-%5Cmathbb%20E%5BY%5D%5E2)
As stated in the question, there is a
probability that any one person will get off at floor 
(here, refers to any of the total floors, not just the top floor). Then the probability that a person will not get off at floor is 
. There are 
people in the elevator, so the probability that not a single one gets off at floor is 
.
So,
which means
![\mathbb E[Y]=\mathbb E\left[\displaystyle\sum_{i=1}^nX_i\right]=\displaystyle\sum_{i=1}^n\mathbb E[X_i]=\sum_{i=1}^n\left(1\cdot\left(1-\dfrac1n\right)^m+0\cdot\left(1-\left(1-\dfrac1n\right)^m\right)](https://tex.z-dn.net/?f=%5Cmathbb%20E%5BY%5D%3D%5Cmathbb%20E%5Cleft%5B%5Cdisplaystyle%5Csum_%7Bi%3D1%7D%5EnX_i%5Cright%5D%3D%5Cdisplaystyle%5Csum_%7Bi%3D1%7D%5En%5Cmathbb%20E%5BX_i%5D%3D%5Csum_%7Bi%3D1%7D%5En%5Cleft%281%5Ccdot%5Cleft%281-%5Cdfrac1n%5Cright%29%5Em%2B0%5Ccdot%5Cleft%281-%5Cleft%281-%5Cdfrac1n%5Cright%29%5Em%5Cright%29)
![\implies\mathbb E[Y]=n\left(1-\dfrac1n\right)^m](https://tex.z-dn.net/?f=%5Cimplies%5Cmathbb%20E%5BY%5D%3Dn%5Cleft%281-%5Cdfrac1n%5Cright%29%5Em)
and
![\mathbb E[Y^2]=\mathbb E\left[\left(\displaystyle\sum_{i=1}^n{X_i}\right)^2\right]=\mathbb E\left[\displaystyle\sum_{i=1}^n{X_i}^2+2\sum_{1\le i](https://tex.z-dn.net/?f=%5Cmathbb%20E%5BY%5E2%5D%3D%5Cmathbb%20E%5Cleft%5B%5Cleft%28%5Cdisplaystyle%5Csum_%7Bi%3D1%7D%5En%7BX_i%7D%5Cright%29%5E2%5Cright%5D%3D%5Cmathbb%20E%5Cleft%5B%5Cdisplaystyle%5Csum_%7Bi%3D1%7D%5En%7BX_i%7D%5E2%2B2%5Csum_%7B1%5Cle%20i%3Cj%7DX_iX_j%5Cright%5D%3D%5Cdisplaystyle%5Csum_%7Bi%3D1%7D%5En%5Cmathbb%20E%5B%7BX_i%7D%5E2%5D%2B2%5Csum_%7B1%5Cle%20i%3Cj%7D%5Cmathbb%20E%5BX_iX_j%5D)
Computing![\mathbb E[{X_i}^2]](https://tex.z-dn.net/?f=%5Cmathbb%20E%5B%7BX_i%7D%5E2%5D)
is trivial since it's the same as ![\mathbb E[X_i]](https://tex.z-dn.net/?f=%5Cmathbb%20E%5BX_i%5D)
. (Do you see why?)
Next, we want to find the expected value of the following random variable, when
:
If
, we don't care; when we compute ![\mathbb E[X_iX_j]](https://tex.z-dn.net/?f=%5Cmathbb%20E%5BX_iX_j%5D)
, the contributing terms will vanish. We only want to see what happens when both floors are not visited.
![\implies\mathbb E[X_iX_j]=\left(1-\dfrac2n\right)^m](https://tex.z-dn.net/?f=%5Cimplies%5Cmathbb%20E%5BX_iX_j%5D%3D%5Cleft%281-%5Cdfrac2n%5Cright%29%5Em)
where we multiply by
because that's how many ways there are of choosing indices 
for 
such that 
.
So,
![\mathrm{Var}[Y]=n\left(1-\dfrac1n\right)^m+2n(n-1)\left(1-\dfrac2n\right)^m-n^2\left(1-\dfrac1n\right)^{2m}](https://tex.z-dn.net/?f=%5Cmathrm%7BVar%7D%5BY%5D%3Dn%5Cleft%281-%5Cdfrac1n%5Cright%29%5Em%2B2n%28n-1%29%5Cleft%281-%5Cdfrac2n%5Cright%29%5Em-n%5E2%5Cleft%281-%5Cdfrac1n%5Cright%29%5E%7B2m%7D)
Let
We want to find
As stated in the question, there is a
So,
which means
and
Computing
Next, we want to find the expected value of the following random variable, when
If
where we multiply by
So,
