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JohanL
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Homework Statement
Insects land in the soup in the manner of a Poisson process with intensity lambda. Insects are green with probability p, independent of the color of the other insects. Show that the arrival of green insects is a Poisson process with intensity p*lambda.
Homework Equations
3. The solutionIm only interested in one step of the solution. Our professor started out calculating the characteristic function of an exponential random variable.
"We solve this by checking that the times, call one such typical time T, between arrivals of new green insects are exp(p*lambda) - distributed"
$$E[e^{j\omega\exp(\lambda)}] = ... = \frac{\lambda}{\lambda - j\omega}$$
Its the next step that is confusing me
$$E[e^{j\omega\exp(T)}] = \sum_{n=1}^\infty (\frac{\lambda}{\lambda - j\omega})^n p(1-p)^n$$
What is he doing here?
I understand that he is dividing the event into a disjoint sum of events and that the characteristic function of a sum of independent variables is the product of the characteristic functions of the variables but still don't understand what the nth power of a characteristic function is doing there. Is the characteristic function of an exponential a waiting time??
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