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I understand how to prove that the sum of $n$ independent Poisson random variables is a poisson random variable mathematically. However, I don't understand this in terms of the concept of Poisson processes.

A poisson process must have accidents occurring at a constant rate, independently and singly. I understand that in the combined process accidents will occur independently and at a rate of the sum of the $n$ different rates but I'm not sure how accidents are guaranteed to occur singly in the combined process. E.g. below, how do you know a white and black dot won't overlap in the merged process?

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Also, if someone can explain how to deduce the rate of the combined process from the concept itself instead of the mathematical proof that would be great.

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  • $\begingroup$ "how do you know a white and black dot won't overlap"? What do you mean by "overlap"? Do you mean they will both be at the same point? $\endgroup$ – Michael Hardy Apr 25 at 0:54
  • $\begingroup$ You stated "I'm not sure how accidents are guaranteed to occur singly" but there is no guarantee that this probability zero event will not occur. Of course, since the probability is zero, this is not a problem. In fact, all that matters is the number of events in a time interval, even if two events happen at the same time. $\endgroup$ – Somos Apr 25 at 1:28
  • $\begingroup$ @MichaelHardy Sorry, I mean: how do you know an accident from the first process doesn't occur at the exact same time as one from the 2nd process in the combined process? And in the diagram, the accidents occurring at the same time would correspond to a black and white dot being at the same point. $\endgroup$ – user898975 Apr 25 at 16:40
  • $\begingroup$ @user898975 : Put the arrival times of one Poisson process on the horizontal axis and of the other on the vertical. Then what is the probability that a point so plotted will be exactly on the diagonal line? This is a continuous distribution in $\mathbb R^2$ and the area of the diagonal line is $0. \qquad$ $\endgroup$ – Michael Hardy Apr 25 at 17:10
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If phone calls arrive at a switchboard at an average rate of $3$ per minute, and at another switchboard at an average rate of $4.2$ per minute, then at the two switchboards combined, they arrive at an average rate of $7.2$ per minute.

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A Poisson Distribution gives the probability of a number of events in an interval generated by a Poisson process. The Poisson distribution is defined by the rate parameter, $\lambda$, which is the expected number of events in the interval: expected here means "on average". It's all about the average number of events in the given time interval. Now you can understand that if you sum two Poisson random variables, with "expected number of events" in a fixed interval given by $\lambda_1$ and $\lambda_2$, then for the sum of them we simply expect a number of events given by the sum of the previous two.

About your drawing, think as if you could write a vertical line at some specific point and count how many points you have on the left of your line. This number is the expected number of events occurring in that particular fixed interval of time. Now you see that it is intuitive to expect, for the sum random variable, "on average", the sum of the number of events occurred in the two processes.

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