# What is Poisson distribution: Definition and 138 Discussions

In probability theory and statistics, the Poisson distribution (; French pronunciation: [pwasɔ̃]), named after French mathematician Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.
For instance, a call center receives an average of 180 calls per hour, 24 hours a day. The calls are independent; receiving one does not change the probability of when the next one will arrive. The number of calls received during any minute has a Poisson probability distribution: the most likely numbers are 2 and 3 but 1 and 4 are also likely and there is a small probability of it being as low as zero and a very small probability it could be 10. Another example is the number of decay events that occur from a radioactive source during a defined observation period.

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1. ### Finding probability related to Poisson and Exponential Distribution

My attempt: (i) ##\lambda =3## (ii) (a) ##P(N_{2} \geq 1=1-P(N_{2} =0)=1-e^{-6} \frac{(-6)^0}{0!}=0.997## (b) ##P(N_{4} \geq 3)=1-P(N_{4} \leq 2)=0.999## (c) ##P(N_{1} \geq 2) = 1-P(N_{4} \leq 1)=0.8## Do I even understand the question correctly for part (i) and (ii)?(iii) The expectation of...
2. ### A Expression of Shot noise when expanding ##a_{\ell m}## coefficients

I would like to arrive at the following expression for the quantity ##o_{\ell}## ( with "DM" for Dark Matter ): ##o_{\ell}=b_{s p}^2 C_{\ell}^{D M}+B_{s p}## with Poisson noise ##B_{s p}=\frac{1}{\bar{n}}(\bar{n}## being the average number of galaxies observed). the index "sp" is for spectro...
3. ### Characterizing random processes

Hello. I would like to kindly request some help with a multi-part problem on identifying random processes as an intro topic from my stats course. I’m fairly uncertain with this topic so I suspect my attempt is mostly incorrect, especially when specifying the parameters, and I would be grateful...
4. ### A LogLikelihood - Poisson distribution

Hello :) I try to fit some parameters of the particle (e.g. energy, direction) be means of log-likelihood minimization. Input data to likelihood function are pulses amplitudes, while Poisson distribution is used. However, the problem is that Poisson distribution is as follows i.e. for higher...
5. ### A Relation between a_{\ell m} noise and Poisson noise with C_{\ell}

We assume two galaxy population, ##\mathrm{A}## and ##\mathrm{B}##; the corresponding maps have the following ##a_{\ell m}## : ## \begin{aligned} &a_{\ell m}^{A}=b_{A} a_{\ell m}^{M}+a_{\ell m}^{p A} \\ &a_{\ell m}^{B}=b_{B} a_{\ell m}^{M}+a_{\ell m}^{p B} \end{aligned} ## Here, ##b_{A}## and...
6. ### Is My Understanding of Poisson Distribution Correct for Different Area Sizes?

Please excuse me for posting in this group.l There seems to be little activity in the statistics group and i might get no response. For this example, lambda, is the average of dogs per 100 square miles = 0.05. if I wanted the probability of 2 dogs in 100 square miles I would calculate.. P(x=2)...
7. ### Using Poisson random variables to calculate this probability

I calculated the mean which is 78.4 And the Standard deviation is 5.6 I thought the answer would be (90^(-78.4)/78.4!)*e^-90 But looking back having a decimal factorial doesn't make sense I have the numerical answers for c)= 0.019226 and d)=0.022750 but I my solution was wrong. Any help on...
8. ### A Analytical expression of Cosmic Variance - Poisson distribution?

I have an expression of Matter Angular power spectrum which can be computed numerically by a simple rectangular integration method (see below). I make appear in this expression the spectroscopic bias ##b_{s p}^{2}## and the Cosmic variance ##N^{C}##. ## \begin{aligned}...
9. ### I Statistic uncertainties in cross section plots - how to calculate?

Hi there, I hope I chose the right forum for my question. So, basically, I'm doing an analysis measuring the number of signal particles in a certain momentum bin i, and doing two corrections: Nsig, i=M*(Nmeas, i-Nbkg, i) Here, M is a matrix covering PID correction and PID efficiencies, and...
10. ### B Poisson distribution having variation coefficient = .5?

Variation coefficient is calculated by And the very definition of poisson distribution is that $$\mu = \sigma$$ So how would any other value but 1 be a possible?
11. ### A Why the Poisson noise level is set to 2 after applying Leahy norm

I am studying about power spectrum analysis in high energy astrophysics. I cannot understand why the Poisson noise level is set to 2 after applying Leahy normalization. $$P_{j}=2 /_{N \mathrm{ph}}\left|a_{j}\right|^{2}$$ The above is the equation for leahy norm, Can I expand the equation from...
12. ### Conditional Probability + Poisson Distribution

Confused and not sure if it is correct, but please do correct my steps. We let event B be that there are at least 3 customers entering in 5 minutes. Hence P(B) = 1- P(X=0)- P(X=1) - P(X=2) = ##1- \dfrac{e^{-5}5^{0}}{0!}-\dfrac{e^{-5}5^{1}}{1!}-\dfrac{e^{-5}5^{2}}{2!} ## = 0.8753... Now we let...
13. ### B Poisson Distribution - Selecting cookies that are indistinguishable

Here's the problem: A chef made 500 cookies randomly mixed with 1000 nuts including 600 almonds and 400 hazelnuts in which each nut is the same size. Suppose the number of pieces of nuts in a piece of cookie follows a Poisson distribution. (a) Suppose cookies are randomly selected one-by-one...
14. ### Poisson distribution and Shot Noise

My setup: I have the an LED (LED370E) in front of a photodiode (S12915-16R). The photodiode is connected to an ADC (DT5751) which has a counting functionality. The way it works is that it counts how many times the signal goes above a certain threshold and makes a histogram out of it. I know...
15. ### A Relationship between Poisson distribution and Poisson Process

Apologies if this has been discussed elsewhere. I know a Poisson process implies a Poisson distribution, but does a Poisson distribution imply a Poisson process? and does the absence of a Poisson distribution imply the absence of a Poisson process? TIA - Sunil
16. ### Poisson distribution probability problem

So I thought you would find the probability of having 0 errors when the mean rate is 1.6. Square that by 5 and multiply that by one minus the probability of having 0 errors to the power of 7. So that is basically the probability of having 0 errors to the power of 5 multiplied by the probability...
17. ### A Deriving a Probability Generating Function for Independent Poisson Variables

Can anyone kindly tell me how I can derive a Probability Generating Function of Poisson Distribution for ##X+Y## where ##X## and ##Y## are independent? I know that PGF for a single variate Poisson Distribution is: ##G(t) = e^{-\lambda (1-t)}##. Then how can I derive a PGF for the same? Is...
18. ### Variance with Poisson distribution

<Moderator's note: Moved from a technical forum and thus no template.> So, I have this problem and I am stuck on a sum. The problem I was given is the following: The probability of a given number n of events (0 ≤ n < ∞) in a counting experiment per time (e.g. radioactive decay events per...
19. ### What Is the Expected Wait Time for the Last Pedestrian at a Traffic Light?

Homework Statement Pedestrians approach to a signal for road crossing in a Poisson manner with arrival rate ##\lambda## per sec. The first pedestrian arriving the signal pushes the button to start time ##T##, and thus we assume his arrival time is ##t=0##, and he always see ##T## wait time. A...
20. ### How Does the PDF of Wait Time Vary for Pedestrians at a Traffic Signal?

Homework Statement Pedestrian are arriving to a signal for crossing road with an arrival rate of ##\lambda## arrivals per minute. Whenever the first Pedestrian arrives at signal, he exactly waits for time ##T##, thus we say the first Pedestrian arrives at time ##0##. When time reaches ##T##...
21. ### Probability and pedestrian wait time density function

Homework Statement Pedestrians approach to a signal at the crossing in a Poisson manner with arrival rate ##\lambda## arrivals per minute. The first pedestrian arriving the signal starts a timer ##T## then waits for time ##T##. A light is flashed after time T, and all waiting pedestrians who...
22. ### A What exactly is a "rare event"? (Poisson point process)

These days I've been reading in the internet about the Poisson Distribution because that was a concept I couldn't manage to understand completely when I studied it, so since then I've been always quite curious about Poisson processes, and how there are a lot of natural phenomena (mostly the...
23. ### [Poisson Stats] Error on half-life for radioactive decay

Hi there, not sure whether this is in the right section but: I've made two runs of a radioactive decay experiment where I've got a log(N) vs. time plots. From this I've got the decay constants and hence the half-life. I've averaged these two half-lives ( = 160 secs) and now I'm trying to work...
24. ### Poisson Distribution -- Rental of a number of television sets

Homework Statement A dealer has a stock of 6 similar television sets which he rents out to customers on a monthly basis. It is known from past experience of the dealer that the monthly demand for the television sets have a Poisson distribution with mean 3.56 (i) Find the probability that in any...
25. T

### Poisson distribution ( approximation)

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 ? Homework EquationsThe Attempt at a Solution This is a...
26. ### A Quantum measurement operators with Poisson distribution

The following is a somewhat mathematical question, but I am interested in using the idea to define a set of quantum measurement operators defined as described in the answer to this post. Question: The Poisson Distribution ##Pr(M|\lambda)## is given by Pr(M|\lambda) =...
27. ### How Does the Poisson Distribution Calculate Tornado Probabilities in Georgia?

Homework Statement The number of tornadoes per year, in Georgia, has a Poisson distribution with a mean of 2.4 tornadoes. Calculate the probability that in any given year, there will be: (i) At most 2 cases. (ii) At least one case. (iii) Calculate the probability that there will be...
28. ### I Poisson distribution regarding expected distance

Hi The question is about diseased trees in an area (Poisson process), and states that λ = 15 diseased trees in a km square. I need to calculate expected distance from a point in the square to a diseased tree. Now I thought that this means that P(diseased tree = 0) ~ Po(15) = 3.059 x 10^-7 Or...
29. ### Performing Metropolis-Hastings algorithm for a Poisson Distribution

Homework Statement The number of busy lines in a trunk group (Erlang system) is given by a truncated Poisson distribution. I am asked to generate values from this distribution by applying the Metropolis-Hastings algorithm. Homework Equations The distribution is given in the attached picture...
30. ### I Is this a Poisson distribution problem?

Hello, I have been thinking about this problem for a while, but I can't decide how it should be tackled statistically. I wonder if you can help, please. Suppose that prostheses for hip replacement are sold mainly by 2 manufacturers, A and B. Since they started being sold 20 years ago, 100 000...
31. ### Calculating Probability using the Poisson Distribution

Homework Statement On average, 2 students per hour come into the class. What is the probability that the time between two consecutive arrivals is in the interval <10 minutes; 50 minutes>. Homework Equations p(k)=P(Y=k)=((lambda*t)k*(e-lambda*t)/k! The Attempt at a Solution I've tried using...
32. ### I Poisson distribution with conditional probability

Hi guys, I have a question about computing conditional probabilities of a Poisson distribution. Say we have a Poisson distribution P(X = x) = e^(−λ)(λx)/(x!) where X is some event. My question is how would we compute P(X > x1 | X > x2), or more specifically P(X> x1 ∩ X > x2) with x1 > x2? I...
33. ### I Proof to the Expression of Poisson Distribution

Hello. Given a range of time in which an event can occur an indefinite number of times, we say a random variable X folows a poisson distribution when it follows this statements: X is the number of times an event occurs in an interval and X can take values 0, 1, 2, … The occurrence of one event...
34. ### I Why do we require conditions for the Poisson Distribution?

Three conditions must be met in order for the Poisson Distribution to be used: 1) The average count rate is constant over time 2) The counts occurring are independent 3) The probability of 2 or more counts occurring in the interval $n$ is zero Simply, why must these conditions be met for valid...
35. ### Radioactive decay, relation between binomial to expon. dist

You can model the probability for radioactive decay as a Poisson distribution. This is the probability for radioactive decay within a specific time interval. (I probably got some of it wrong here). P(k,μ)=λ^k⋅exp(-μ)/k! Is there a way to use this formula to derive the other formula for...
36. ### I Poisson process approximation error

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...
37. ### MHB Poisson Distribution Problem

Hello all, I have this Poisson distribution question, which I find slightly tricky, and I'll explain why. The number of car accidents in a city has a Poisson distribution. In March the number was 150, in April 120, in May 110 and in June 120. Eight days are being chosen by random, not...
38. ### Poisson Distrib.: Estimating Mean of Data Set

Homework Statement I am given a data set known to come from a poisson distribution. Homework Equations Poisson distribution The Attempt at a Solution I want to calculate the mean of the data set for use in the Poisson Distribution function. How do I best estimate this. Do I take the...
39. ### How Do Poisson and Binomial Distributions Apply to Wire Flaw Analysis?

Homework Statement We assume that the number of structural flaws on a long wire have obey Poisson distribution law. On average we find 1 flaw every 5 meters. a) What is the probability that a 20 m long section will have maximum 2 flaws? b) We slice the wire into 1 m long sections. What is the...
40. ### Solving Poisson Distribution Problems: Questions on Calls/Minute

Homework Statement A telephone operator receives four phone calls in three minutes on the average. Let a Poisson random number X denote the number of phone calls per minute to this operator. (a) Find the probability that this operator receives two phone calls in a minute. (b) Find the...
41. ### Binomial -> Poisson distribution question

Homework Statement A teacher has an infinite flow of papers to mark. They appear in his office at random times, at an average rate of 10 a day. On average 10% of the manuscripts are free from errors. What is the probability that the teacher will see exactly one error-free manuscript (a) after...
42. ### When/how to reject Poisson distribution hypothesis?

Homework Statement I have run into a situation that my gut tells me is impossible (alright extremely unlikely) when assuming a Poisson distribution. I want to make this gut feeling more formal by testing it against a Poisson distribution. Sadly I'm not a schooled statistician. Generalised...
43. ### Exercise on Poisson distribution

Homework Statement An experimenter measures the counting rate from a radioactive source as 10,150 counts in 100 minutes. Without changing any of the conditions, the experimenter counts for one minute. There is a probability of about 15 percent that the number of counts recorded will be fewer...
44. ### 52 card Poisson Distribution experiment?

Hi, I was trying to think of a way to generate a Poisson distribution using a single deck of 52. Say I am looking at the position of the Ace of spades in the deck after a number of shuffle rounds (1 shuffle round is 7 riffle type shuffles). Success is that an Ace of spades is on top of the...
45. ### Looking for a modified Poisson distribution

I'm looking to model a system in which events are nearly perfectly randomly distributed but with a slight tendency for events to avoid each other. As you know, if the system were perfectly random, I could use a Poisson distribution. The probability distribution for the number of events would...
46. ### MHB Wrecking my brains on this Poisson Distribution question.

Hi. I normally can solve poisson distribution questions with ease. But this one question had me thinking for hours on end with no solution. It would be great if someone can help me. QN: The number of incoming calls per minute, X, to a telephone exchange has a Poisson distribution with mean 2...
47. ### MHB What is the Probability of Getting a Raise at McBurger's Drive-Thru?

Problem: McBurger’s drive-thru has only one service window and serves an average of 2 customers every 5 minutes. 70% of customers order drinks from the drive-thru. The manager monitors the employee at the drive-thru for the next 3 hours. He will give the employee a raise if exactly 20 customers...
48. ### Stats - mle poisson distribution -- quick question

This is probably a stupid question , but, It's easy enough to show that the mle of a poission distribution is ## \bar{x}##: ## \hat{ \lambda}= \bar{x} ## But,I'm then looking at the generalized ratio test section of my book, multinomial, it esitmates ## \lambda ## for some data by ## \sum...
49. ### Solving Poisson Distribution: Part IV - Tank of Water (10^5 cm3)

Homework Statement i am having problem with part iv ) . the ans is 0.04519 . can anyone tell me how to do this ? i have solved part i , ii and iii ..p/s line 1: A tank contain 10^5 cm3 of water Homework Equations The Attempt at a Solution
50. ### Poisson distribution involving 2 variables

Homework Statement A coffee shop sell tea and coffee. The number of cups of coffee sold in a minute can be assumed to be a random poisson variable with mean = 1.5 . The number of cups of tea sold can be assumed to be an independent random variable with mean = 0.5. Calculate the probablity...