Poisson process approximation error

In summary, Sal defines success as 1 or more cars passing in a minute and failure as 2 or less cars passing. If you reduce the time interval to 3600 seconds, you will never get more than one car passing in a minute.
  • #1
CynicusRex
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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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  • #2
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
TheBlackAdder said:
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
Stephen Tashi said:
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
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)
 

1. What is a Poisson process approximation error?

A Poisson process approximation error is the difference between the actual number of occurrences in a given time interval and the predicted number of occurrences based on a Poisson distribution. It measures how well the Poisson distribution approximates a real-life event or phenomenon.

2. How is Poisson process approximation error calculated?

Poisson process approximation error is typically calculated using the formula: E = √(λ), where λ is the average rate of occurrence of the event. This formula assumes that the average rate is constant over time and that the event occurrences are independent of each other.

3. What factors can affect Poisson process approximation error?

Poisson process approximation error can be affected by various factors such as the time interval being considered, the total number of events, and the average rate of occurrence. Other factors that can impact the accuracy of the approximation include the assumption of independence between event occurrences and the presence of any external influences on the event.

4. Why is Poisson process approximation important in science?

Poisson process approximation is important in science because many real-world phenomena can be modeled using a Poisson distribution. By understanding the error associated with this approximation, scientists can better analyze and interpret data, make predictions, and identify any potential limitations of their models.

5. How can Poisson process approximation error be minimized?

To reduce Poisson process approximation error, scientists can increase the number of events observed, decrease the time interval being considered, or improve the accuracy of the average rate of occurrence. Additionally, considering other probability distributions or incorporating external factors into the model can also help minimize the error.

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