Poisson distribution ( approximation)

• tzx9633
I think there are a few reasons why people use spreadsheets for this kind of calculation:1. Simplicity: Many people are not familiar with coding or using software like Matlab or Mathematica, but they are familiar with spreadsheets. Spreadsheets are also easily accessible and commonly used in various industries and fields, making them a convenient option for many.2. Familiarity: Spreadsheets are often used for data analysis and calculation in fields like finance, accounting, and business, so people who work in these fields are already familiar with using them. It may be easier for them to stick with what they know rather than learning a new software.3. Flexibility: Spreadsheets allow for a lot of flexibility in formatting and
tzx9633

Homework Statement

The number of flaws in a plastic panel used in the interior of cars has a mean of 2.2 flaws per square meter of panel .
What's the probability that there are less than 20 surface flaws in 10 square meter of panel ?

The Attempt at a Solution

This is a poisson distribution problem , am i right ?

X ~Po(22) for 10 square meter of panel

It's quite insane to calculate the probability from 1 to 20 , right ? I'm wondering is there any appoximation method so that i can solve this question easily ?

tzx9633 said:

Homework Statement

The number of flaws in a plastic panel used in the interior of cars has a mean of 2.2 flaws per square meter of panel .
What's the probability that there are less than 20 surface flaws in 10 square meter of panel ?

The Attempt at a Solution

This is a poisson distribution problem , am i right ?

X ~Po(22) for 10 square meter of panel

It's quite insane to calculate the probability from 1 to 20 , right ? I'm wondering is there any appoximation method so that i can solve this question easily ?
You didn't show the solution, but I'm presuming you know that it is ## P=e^{-\lambda} \, \sum\limits_{k=o}^{19} \frac{\lambda^k}{k!} ## where ## \lambda =10(2.2)=22 ##. The only way I know of to get this answer is by letting the computer process it. Perhaps someone else knows some approximation method. Meanwhile, with today's spreadsheets, this sum is readily computed. ## \\ ## Because the mean ## \lambda=22 ##, I anticipate the numerical answer is going to come out somewhere near ## P=\frac{1}{2} ##. If you process it by computer, let us know what you come up with. :)

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tzx9633 said:

Homework Statement

The number of flaws in a plastic panel used in the interior of cars has a mean of 2.2 flaws per square meter of panel .
What's the probability that there are less than 20 surface flaws in 10 square meter of panel ?

The Attempt at a Solution

This is a poisson distribution problem , am i right ?

X ~Po(22) for 10 square meter of panel

It's quite insane to calculate the probability from 1 to 20 , right ? I'm wondering is there any appoximation method so that i can solve this question easily ?

Whether or not it is a Poisson probability problem depends a lot on details of the manufacturing process that we have not been told. If the positions and numbers of flaws are, indeed, random and independent, then the Poisson model is justified. (Often, textbooks will give problems like this one, where crucial information is lacking, in the expectation that the student will make some assumptions because of context---such as the problem occurring in a chapter about the Poisson distribution, for example.)

Since the mean (22) is not very small, one can try a Normal approximation to the Poisson; then if you have a scientific calculator with the Normal distribution on it you could use that to get an answer that is, at least, in the right "ballpark". However, it would not be particularly accurate, even if you included a so-called "1/2" correction.

With modern tools (spreadsheets,etc.) computing and summing 20 Poisson probabilities is a snap, and including all of them is far from "insane". Of course, on an exam the situation would be different, and making a reasonable approximation would be more important.

You need to tell us what the distribution is. It certainly feels like it could be Poisson.

-- edit: I basically concur with @Ray Vickson's post. However I've thrown in a few different ways of estimating / bounding the results that may be helpful. ---

I'd suggest solving this problem a few different ways, and then comparing results at the end.

tzx9633 said:
X ~Po(22) for 10 square meter of panel

It's quite insane to calculate the probability from 1 to 20 , right ? I'm wondering is there any appoximation method so that i can solve this question easily ?

What is the ##\lambda## i.e. key parameter, of this Poisson distribution?

1.) It is not insane, and you should in fact go from 0 to 19. (Or solve for the complement that goes from ##\{20, 21, 22, 23, ...\}##.) This is one way, and you should try it in excel or with something like python or matlab.

2.) How would you tackle the complement? You could come up with your own approach if you have any knowledge of how to bound the power series for the exponential function. This could be worth playing around with, or not.

3.) You may also consider using a normal approximation to a poisson.

4.) I think the process at whatever scale gets a fresh start, as Poisson's do. You could select this as some large n discrete (read: Bernouli trials) events, as Poisson's can be interpreted as a limit of a Bernouli Process. Select some appropriately large n (in fact try a few different big values for n). It takes some care to do this right and properly convert between large Bernoulis and Poissons, but it isn't that tough. The model is you have a simple binary outcomes where something is either flawed or not, and we have independence in here between bernouli trials. This gives nice and easy approach, using Chernoff Bounds. The most accessible writeup I've seen is in section 20.6.2 (i.e. page 891) of MIT's 'Math for CS', available here: https://courses.csail.mit.edu/6.042/spring17/mcs.pdf

22 is a rather large number. If you just want a reasonable approximation, apply the central limit theorem.

To be honest, I never understood why people use spreadsheets when the corresponding code in Matlab or Mathematica is one line.

Orodruin said:
22 is a rather large number. If you just want a reasonable approximation, apply the central limit theorem.To be honest, I never understood why people use spreadsheets when the corresponding code in Matlab or Mathematica is one line.
I am not terribly computer savvy. (Probably an understatement). I would normally compute this with an EXCEL spreadsheet, but I don't even have that capability on my present computer. :)

I am not terribly computer savvy. (Probably an understatement). I would normally compute this with an EXCEL spreadsheet, but I don't even have that capability on my present computer. :)
I am not complaining about you in particular, just that many people use spreadsheets when a simple Matlab or Mathematica command not only would have been easier to write down, but also would have given a more accurate result.

Example: A student I am familiar with was asked to compute an integral did it with Mathematica in one line and was told by the professor that that was an unnecessarily complicated way of doing things and "we just want something easy that works". The professor's solution involved computing the integrand at 20 different points in a spreadsheet and summing them ... Of course, what Matlab or Mathematica would do would be exactly the same, but with a much larger number of points and with a much less cumbersome input. Doing stuff like this in a spreadsheet is like rewriting the integration routines, just in a much less suited language and with less precision ...

Orodruin said:
I am not complaining about you in particular, just that many people use spreadsheets when a simple Matlab or Mathematica command not only would have been easier to write down, but also would have given a more accurate result.

Example: A student I am familiar with was asked to compute an integral did it with Mathematica in one line and was told by the professor that that was an unnecessarily complicated way of doing things and "we just want something easy that works". The professor's solution involved computing the integrand at 20 different points in a spreadsheet and summing them ... Of course, what Matlab or Mathematica would do would be exactly the same, but with a much larger number of points and with a much less cumbersome input. Doing stuff like this in a spreadsheet is like rewriting the integration routines, just in a much less suited language and with less precision ...
I know=I am retired=today's generation has somewhat better tools than what some of us had. I, myself, like doing calculations by hand whenever possible, and this is a skill that I think the present generation, for the most part, does not do quite as well as my generation. :)

Orodruin said:
22 is a rather large number. If you just want a reasonable approximation, apply the central limit theorem.To be honest, I never understood why people use spreadsheets when the corresponding code in Matlab or Mathematica is one line.

When I composed my response I included a note about on-line Poisson calculators, but for some reason it did not come through in the final, posted version. Anyway, I mentioned spreadsheets---although I mostly try to avoid them myself---because they are likely to be the most widely available tools on almost any computer a student will own. Personally, I just slap such calculations into Maple and let that take care of things. (As for spreadsheets, I started using them primarily because the introductory textbooks in my subject---Operations Research---went almost exclusively the spreadsheet route. I never liked it, but went with the flow.)

Back in the Stone Age when I was a student--and even in more recent times when I started teaching---I would go to the library and consult statistical tables if I did not happen to have the appropriate ones available in the back of some book. Writing a Fortran program on punch cards would be more trouble than it was worth for a problem of this type; and doing things by hand using a sliderule and/or log tables also seems a bit excessive. That was when I really learned the importance of making reasonable approximations.

Perhaps it is worth mentioning to the OP that if a Gaussian (normal distribution) approximation is used for this problem, the mean ## \mu=\lambda ##, and ## \sigma=\sqrt{\mu}=\sqrt{\lambda} ##. (Compare to binomial, where ## \sigma=\sqrt{Npq} ##). That gives a ## z=(20-22)/4.7=-.425 ##. The table showed this gives ## P=.335 ##. It would be interesting to hear what the OP got when he performed the sum.

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Ray Vickson said:
Whether or not it is a Poisson probability problem depends a lot on details of the manufacturing process that we have not been told. If the positions and numbers of flaws are, indeed, random and independent, then the Poisson model is justified. (Often, textbooks will give problems like this one, where crucial information is lacking, in the expectation that the student will make some assumptions because of context---such as the problem occurring in a chapter about the Poisson distribution, for example.)

Since the mean (22) is not very small, one can try a Normal approximation to the Poisson; then if you have a scientific calculator with the Normal distribution on it you could use that to get an answer that is, at least, in the right "ballpark". However, it would not be particularly accurate, even if you included a so-called "1/2" correction.

With modern tools (spreadsheets,etc.) computing and summing 20 Poisson probabilities is a snap, and including all of them is far from "insane". Of course, on an exam the situation would be different, and making a reasonable approximation would be more important.
I have learned normal and poisson approximation to binomial , but not normal distribution approximation to poisson distribution , does the normal distribution approximation to poisson distribution exist ? Sorry

tzx9633 said:
I have learned normal and poisson approximation to binomial , but not normal distribution approximation to poisson distribution , does the normal distribution approximation to poisson distribution exist ? Sorry

Yes. For large ##\lambda##, ##\text{Poisson}(\lambda)## and the ##\text{Normal}(\lambda, \sqrt{\lambda})## are quite close. In your case the issue is whether or not '22' is a large enough number to justify the replacement. The basic reason here is that your ##X \sim \text{Po}(22)## can be considered as a sum of ##N = 22## independent, identically-distributed random variables, each having distribution ##\text{Po}(1)## (or as a sum of ##M = 20## variables having distribution ##\text{Po}(1.1)##.) Anyway, you are getting into the territory of summing a moderate-to-large number of iid random variables, so are getting close to the territory where the Central Limit Theorem applies.

See, eg.,
http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Limits_Norm2Poisson
or
http://www.socr.ucla.edu/Applets.dir/NormalApprox2PoissonApplet.html

What is Poisson distribution and when is it used?

Poisson distribution is a statistical distribution that is used to model the probability of a certain number of events occurring in a given time interval or space when the events are independent of each other and the average rate of occurrence is known. It is typically used in situations where the number of events is rare and random, such as in radioactive decay, traffic accidents, or customer arrivals.

What is the formula for Poisson distribution?

The formula for Poisson distribution is P(x; λ) = (e^-λ * λ^x) / x!, where x is the number of events, λ is the average rate of occurrence, and e is the mathematical constant approximately equal to 2.71828. This formula calculates the probability of x events occurring in a given time interval or space.

How is Poisson distribution different from other probability distributions?

Poisson distribution is different from other probability distributions in that it is used to model rare and random events, while other distributions may be used for more common events. It also assumes that the events are independent of each other, which is not always the case in other distributions.

Can Poisson distribution be used to approximate other distributions?

Yes, Poisson distribution can be used to approximate other distributions such as binomial distribution when the number of trials is large and the probability of success is small. It can also be used to approximate normal distribution when the average rate of occurrence is large.

What are the limitations of Poisson distribution?

One limitation of Poisson distribution is that it assumes the events are independent of each other, which may not always be the case in real-world situations. It also assumes a constant average rate of occurrence, which may not be true in all cases. Additionally, Poisson distribution is only applicable to discrete data, so it cannot be used for continuous data.

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