Poisson arrivals and mgf

[Gekko]

Homework Statement

Suppose cars arriving at an intersection follow a poisson distribution and that the number of passengers in the nth car is Yn where n>=1 and they are iid and independent of Nt each with moment generating function My(k)

Write Zt as a sum of iid random variables and show:

mgf of Zt = exp[ lambda*t*(My(k)-1)]

The Attempt at a Solution

Probability of Yn passengers in the nth car is P(Yn) * P(Nt)

(1/n) * [exp(-labmda*t) *(lambda*t)^n] / n!

If I calculate the mgf from here it doesn't lead to the correct answer.

How is Zt written as a sum of iid random variables from this?
Thanks in advance for any help

 

[mighty2000]

or guidance you can provide.
Thank you for your post. I am a scientist and I would like to offer my assistance in solving this problem.

To write Zt as a sum of iid random variables, we need to consider the number of cars that arrive at the intersection in a given time interval. Let's call this number Nt. We know that Nt follows a Poisson distribution with parameter lambda*t, where lambda is the arrival rate of cars per unit time.

Now, for each car that arrives, we have the number of passengers Yn, which follows its own distribution with a moment generating function of My(k). Since Yn is independent of Nt, we can write Zt as the sum of Yn for each car that arrives, i.e. Zt = Y1 + Y2 + ... + YNt.

Since Yn are iid (independent and identically distributed) random variables, we can use the property that the moment generating function of a sum of iid random variables is the product of their individual moment generating functions. Therefore, the moment generating function of Zt is:

Mz(t) = M(Y1 + Y2 + ... + YNt) = M(Y1)*M(Y2)*...*M(YNt)

= [My(k)]^Nt

= [My(k)]^(lambda*t)

= exp[lambda*t*(My(k)-1)]

Hence, the moment generating function of Zt is exp[lambda*t*(My(k)-1)].

I hope this helps. Let me know if you have any further questions.