Question

The number of people who enter an elevator on the ground floor is a Poisson random variable with mean 10. If there are N floors above the ground floor and if each person is equally likely to get off at any one of these N floors, independently of where the others get off, compute the expected number of stops that the elevator will make before discharging all of its passengers.

Answer 1

Given Information:

Distribution = Poisson

Mean = 10

Required Information:

Expected number of stops = ?

Answer:

`E(N) = N(1 - e^(-0.1N) )`

Explanation:

The number of people entering on the elevator is a Poisson random variable.

There are N floors and we want to find out the expected number of stops that the elevator will make before discharging all of its passengers.

Mean = μ = 10

The expected number of stops is given by

`E(N) = N(1 - e^(-mN) )`

Where m is the decay rate and is given by

Decay rate = m = 1 /μ = 1/10 = 0.10

Therefore, the expected number of stops is

`E(N) = N(1 - e^(-0.1N) )`

If we know the number of floor (N) then we can the corresponding expected value.

Answer 2

Expected no of stops=N(1-exp(-10/N)

Explanation:

Using equation

X=∑
`I_(m)`

where m lies from 1 to N

solve equation below

E(X)=N(1-exp(-10/N)