Stochastic Processes, Poisson Process | Expected value of a sum of functions.

[dharavsolanki]

Homework Statement

Suppose that passengers arrive at a train terminal according to a poisson process with rate "$". The train dispatches at a time t. Find the expected sum of the waiting times of all those that enter the train.

Homework Equations

F[X(t+s)-X(s)=n]=((($t)^n)/n!)e^(-$t))

It is the equation of Poisson Processes.

The Attempt at a Solution

The waiting time for every person is unique, since he arrives at a different time. So, the sum of the waiting time will have a certain value. However, I am unable to understand how can the sum have an "expected value". I mean, what parameters is the sum depending on? I can only see time as a variable here. Ofcourse, the other variable is the number of people arriving, but once that is set up, shouldn't the sum be unique?


Can you help me out in setting up the problem? I am sure that If I am given a setup and equations, I can arry out the solutions myself.

Thank you.

 

[dharavsolanki]

Hey, is this problem really so difficult that there's been no attempt at it yet! I need it kinda urgently. Please help!

 

[rootX]

I will try to give it a shot:Expected time for n=0 person: 0*((($t)^n)/n!)e^(-$t)) = 0
Expected time for n=1 person: 1*((($t)^1)/1!)e^(-$t)) <-- just a constant value so expectation of this just equals this
...
Expected time for n=k person: k*((($t)^k)/k!)e^(-$t))

E_total = E[@n=0 + @n= 1 + ...+ @n=infinity]
= E[@n=0]+...+E[@n=infinity]

And you will get
sum (as k from 0 to inf) k*((($t)^k)/k!)e^(-$t))