Poisson process approximation error

  • #1
TheBlackAdder
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X = # of cars that pass in one hour
E(X) = λ = n * p
λ cars/1hour = 60min/hour * (λ/60) cars/min

In this old video (5:09) on poisson process Sal asks: "What if more than one car passes in a minute?"
"We call it a success if one car passes in one minute, but even if 5 cars pass, it counts as 1 car."

I don't understand the statement in bold. Why do 5 cars count as 1 car?

I do understand that λ/60 = the probability that a car passes in a minute (the probability of success in a minute) and that we make it more accurate if we would again decrease the interval to 3600 seconds instead of 60 minutes.

As I wrote this, I think I got it by typing "interval". Does it count the event of '5 cars in one minute' as '1 car in one minute' because the one minute interval limits the interpretation of success to "car/minute = success"?
(As in: as long as 0<cars = success, and then discards the actual number of cars which passed)

Sorry for the likely unclear explanation, but I was/am terribly confused by this probably obvious fact.
 
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Answers and Replies

  • #2
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Your lamda defines it as 60 cars/ min. I imagine the reason he said this is sticking to that definition so even in the situation where you have more than 1 car pass within a minute you refer back to what you defined as lamda and the logic holds.
 
  • #3
Stephen Tashi
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In this old video (5:09) on poisson process Sal asks: "What if more than one car passes in a minute?"
"We call it a success if one car passes in one minute, but even if 5 cars pass, it counts as 1 car."

I don't understand the statement in bold. Why do 5 cars count as 1 car?

.
He is using a binomial distribution to approximate the situation. A binomially distributed random variable counts the number of successes per N trials. In this case, each trial is a 1 minute interval. He had to define what a "success" is. He isn't allowed to define various "degrees of success". To do that, he'd have to use a different distribution.

We can ask why he didn't define "success" as something like "3 or more cars pass" and "failure" as "2 or less car pass". The reason he doesn't do that is that he is showing how a poission distribution arises as the limit of using binomial distributions defined on smaller and smaller time intervals. One of the assumptions of the poission process is that in very small time intervals there is almost no chance that more than one "success" will happen. So, in a poission process, we don't have the possibility that two or more cars pass us "at the same time". For purposes of teaching, it is more natural to define "success" as "1 or more cars pass in the interval" and then to argue that if you use a very small interval that you will never get more than one car passing in that interval. That approach only involves reducing the time interval. It doesn't involve changing the definition of "success".
 
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  • #4
TheBlackAdder
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He is using a binomial distribution to approximate the situation. A binomially distributed random variable counts the number of successes per N trials.
You mean n trials? (Only time I can recall N being used is as population size.)

PS I'm 99% confident that I'm 95% sure that I somewhat understand now. Thank you.
 
  • #5
TheBlackAdder
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Nvm, I found N is total sample size and n part of that sample.
(Can't seem to edit my last post but I wanted to clear that up)
 

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