Questions tagged [poisson-distribution]

A discrete distribution defined on the non-negative integers that has the property that the mean is equal to the variance.

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Confusion on units for the Poisson distribution when it is used to model variables with units

This question stems from the comment section of this question: Bus wait time under Poisson distribution, where it seems that The properties of the Poisson don't make sense for times because the units ...
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Bus wait time under Poisson distribution

A bus will depart every 10 minutes from the origin, and the time it takes to travel to station $A$ follows a Poisson distribution with expectation of 10 minutes. Alice arrives at station $A$ around 9:...
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What is the conditional expectation of two random Poisson variables?

Say I have two random variables $Z^{p}_{i} \mid Y^{p}_{i} \sim \text{Poisson} \left(t^{p}_{i}Y^{p}_{i}\right)$ and $Z^{q}_{i} \mid Y^{q}_{i} \sim \text{Poisson} \left(t^{q}_{i}Y^{q}_{i}\right)$ for ...
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How to interpret pvalue for Chi² goodness-of-fit test for Poisson distribution

I'm a bit confused about the pvalues I'd get when checking my own data against a theoretical Poisson distribution. Following e.g. this post here: https://stats.stackexchange.com/a/78175/66544 I'd ...
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What variable type to choose for a Poisson distributed variable if software package only allows for nominal, ordinal or continuous

In the area of cluster analysis I want to calculate the dissimilarity for my data (actual use case is to feed in the dissimilarities into a plotting function to calculate so called silhouette plots). ...
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Regression left limited dependet variable

My scope is to analyze the impact of certain variables on the change in sales. As you can see, my dependent variable is a proportion of two variables and is limited to -1 (-100%). On the other hand, ...
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Obtaining formulae for Poisson confidence interval

I have a sample $X=(X_1,\dots,X_n)$ of $i.i.d$ Poisson variables such that $n=100,\overline{X}=8.8$. My goal is to obtain a $80\%$ confidence interval for the parameter $\lambda=\theta$. That is, the ...
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$Y_1, Y_2, ... $ are iid Poisson($\lambda$)

We observe $Y_1, Y_2, ..., Y_T$ such that $T$ is the first $t\geq 1$ for which $Y_t>0$. Define $Y=Y_T$ (a) Find MLE $\hat{\lambda}$. (b) What is the relative bias, $[E(\hat{\lambda})- \lambda]/\...
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Log-likelihood Gaussian cox process

I have a question related to the slide below. How do you obtain the log-likelihood function? Isn't it equal to: $$ \displaystyle \sum_{x,t} \log\left( \int_{-\infty}^{+\infty} f(D_{x,t} = d_{x,t} \mid ...
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How does the result $\dfrac{1}{n^T} \dfrac{T!}{\prod_{i = 1}^n Y_i!}$ tell us what distribution $T(\mathbf{Y})$ is?

This follows on from my question here. I have the following problem: Let $Y_1, \dots, Y_n$ be a random sample from a Poisson distribution $\text{Pois}(\lambda)$. Recall, the $\text{Pois}(\lambda)$ ...
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limit of $\frac{\lambda}{\chi _{2Y}^{2}}$ as $Y \sim Poisson(n\lambda)$ and $𝜆 → ∞$

There are the following lines in Casella & Berger on page 438, before the equation (9.2.22): ..., write $$\lambda = \frac{\lambda}{\chi _{2Y}^{2}}\chi _{2Y}^{2}$$ where $\chi _{2Y}^{2}$ is a chi ...
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Statistical uncertainty on an Poisson distributed value estimated using Poisson distributed variables

Let's say I have the number of events in four regions in phase space. Where the regions are named: $A, B, C$, and $D$. These are Poisson distributed and not correlated with one another, and the number ...
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Under what constraints , if any, does the binomial distribution become equal to the normal distribution? [duplicate]

I understand that when n approaches infinity binomial distribution also approaches a Poisson distribution. What about the normal distribution? I googled and found that if n approaches infinity and p ...
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which type of model to predict percentages with an almost bi modal distribution

I wonder, which type of regression (?) model I could use for a dependent variable corresponding to percentages with an almost bi modal distribution like this? The percentages correspond to weekly room ...
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Winbugs/OpenBUGS code example for Poisson rates?

My data comes from 12 regions. I have the total number of events and the total person-time for each region. For example: ...
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Poisson distribution transformation

I'm quite new to biostatistics so I apologize if my question is too dumb. I'm studying data transformation in biostatistics to fit my data to the normal distribution. I started with the Poisson ...
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Best estimate of a variable from two observations with different efficiencies

Problem Suppose to measure the frequency of a certain rare event (e.g. particle count) with two instruments $I_1$ and $I_2$ for a time $\bar{t}$, the same for both instruments. We expect the same ...
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Method of moments estimator for a probability of an event

I need to find the method of moments estimator for $P(pois(\lambda)=0)$. I already worked out the MME $\hat{\lambda}=\bar{X}$ but I'm not sure how to proceed here because I can only see how this ...
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Specifying distribution in GBM

In the GBM package, we can specify which distribution to use that represents our response variable. I have count data and usually, we specify the distribution as Poisson for count data. But, when I ...
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Meteorites and Poisson parameter estimation

I heard that the number of meteorites of the given size that hit the Earth follows the Poisson distribution. I am wondering how to estimate the Poisson parameter $\lambda$ and its 95% c.i. if I have ...
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Infer (supposed) Poissonian probability from data

Suppose to count the drops of rain in a square meter in 15 seconds, producing 16 observations: 40, 20, 24, 15, 23, 12, 39, 26, 29, 33, 16, 36, 17, 32, 40, 15. What is the probability of counting 28 ...
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I want to make sure the use of Poisson count model OR negative binomial model is right for my study

I have a planned study to conduct in few weeks. As it is my first time to deal with a 'count' dependent variable, I have been searching on this website what statistics tools I should use. I have two ...
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What is $E[X]$ and $\text{Var}(X)$ if $X$ follows $Pois(T^2)$ and $T$ follows Exponential distribution [duplicate]

I'm new to this community. I have problem in finding expected value and variance of R.V.s that are composed of other R.V.s following other distributions. Suppose $X \sim Pois(T^2)$ where $T \sim Exp(\...
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In a zero-inflated Poisson model, what is the relationship between the rate, the 0-probability, and dispersion?

Suppose a random variable $Y$ is distributed as a two-parameter zero-inflated Poisson. That is, $Y=XW$ with $X \sim \text{Poisson}(\theta)$ and $W \sim \text{Bernoulli}(\nu)$. Suppose I obtain a ...
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Intersection of multiple gamma distributions

Let's say I own a few hundred McDonalds locations. In a subset of those (say 100) I observe vegans eating there and I estimate the arrival time of vegans at these 100 restaurants using a Poisson ...
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Law of Total Variance

I trying to experiment with law of total variance in order to empirically recreate theoretical results. In particular I am interested in verifying that: $$ Var(Y) = E[Var(Y|X)] + Var(E[Y|X]) $$ Let's ...
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Improving the estimate by adding a new data point

Background: Let say we have a model $f(x)\approx k(1+x)$, for some function $f$ (e.g. received power) along the dimension $x$ (e.g. horizontal axis), where $k$ is an unknown scalar (or rather not a ...
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Native method to calculate confidence upper bounds for the mean of a Poisson Distribution

Reading about how to calculate confidence intervals for Poisson distributions, one says that if the unknown mean is high you can use the Normal distribution approximation and use $\widehat\lambda\pm c\...
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Flipping Variables for a Poisson Distribution

Not sure exactly how to word this, but I'm stuck on a current question about poisson distributions. So far, the questions have gone along the lines of "What is the probability that x SUCCESSES ...
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Probability of a discrete variable depending on continuous variables

I have one random variable : $Y$ and a set of three parameters $\vec{X}=(X_1,X_2,X_3)$. The variable $Y$ is discrete. I don't know its distribution and I am trying to extract it from data. I am doing ...
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How to calculate the average probability of a poisson distribution?

My Poisson variable is that people trip at a rate of 1.2 people per 12 hours. I was able to find that the probability at least two people trip a day is 0.2613 since I just did $P(X=2)=(e^{-2.4})$ $(2....
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If a Poisson and can be approximated by a Normal, can the average of Poissons be approximate by the average of Normals?

If I have $n$ variables $X_1, \ldots, X_n$ where each $X_i\sim\mathcal{P}(\lambda)$, then $\sum_{i=1}^n X_i\sim\mathcal{P}(n\lambda)$ (I use $\mathcal{P}$ to mean Poisson distribution). However, $\...
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How do I overlap a Poisson distribution with a histogram

I expect to get a Poisson distribution over my histogram but as I can see from the graph, I get a straight line. Would really appreciate insight.
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Contradictory behaviour on sums of poisson variables

Trying to solve this problem of mine (you don't actually need to read the linked problem, the problem is rephrased below with a different assumption): Estimating a sample size such that its sum has ...
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Measuring unusual death [closed]

Given the Prussian Horse Data here: https://www.randomservices.org/random/data/HorseKicks.html Is there a way to find out which corp has an unusually high number of deaths? (Note that Prussian horse ...
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Estimating a sample size such that its sum has some probability of not crossing some upper bound

We have a random variable $X\sim Pois(\lambda)$ with $\lambda$ unknown, and given any random sample $S=\{s_1, \ldots, s_{|S|}\}$ generated from $X$, we define $$\text{numsum}(S)=\sum_{i=1}^{|S|}s_i$$. ...
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Does a constant offset term meaningfully influence Poisson GLM?

If my offset term is a constant value, does this change the outcome of a Poisson GLM regression? I have survey count data that we summarize as a rate (Counts/Area-Surveyed). In this case, transect ...
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Using GLM: Gaussian, Poisson vs Gamma

I am trying to perform a GLM analaysis using R for an outcome that is: Bounded by 0 - 10 In steps of 1 (Numerical Rating Scale for Pain: 0 - 10) I have a set of demographic factors, age, sex etc, ...
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Sampling from an infinite population

SAMPLE SIZE I am sampling astronomical data and measuring clusters of galaxies (within a range of 100 megaprsecs) as nodes in a complex network, where k is a metric ...
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Identify Stock Outages Using Only Sales Data

I have a dataset of sales data for a retailer for a number of SKUs. I'm trying to use this data to identify when a particular SKU was out of stock. I do not have purchase order data (the retailers ...
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Using Julian Day and Count Data in the same GLM

I have count data in a poisson distribution. My goal is to determine whether the 5 habitats surveyed (predictor variable) differ in their number of individuals (animals) (the response variable). I am ...
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New distribution when Negative Binomial dispersion parameter is taken to be 1

What distribution do you arrive at if the dispersion parameter is taken to be 1 in Negative Binomial Distribution?
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Conditional probability mass function of number of Poisson random variable given their sum values

We have a discrete random variable $N$, and $X_1, X_2, ... X_N$ are i.i.d Poisson random variables with parameter $\lambda$. Denote $Y = \sum_{i=1}^{N} X_i$. What I want to know is: If finding the ...
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Probability distribution & Interpretation

If my distribution looks similar to Poisson but not an actual Poisson distribution which is verified using a QQ plot, is there any way to convert such distribution to a proper Poisson distribution ...
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Assumptions of compound Poisson model

My understanding of a compound Poisson RV is one defined as $$Y=\sum_{n=1}^N X_n$$ where $\{X_n\}_{n\in\mathbb{N}}$ is a sequence of identically distributed and mutually independent (iid) RVs $N$ is ...
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Poisson or Binomial distribution for modeling?

I want to model the change in rate of an event in a sampled population. The populations are grouped according to some grouping variable. Each individual observation is of one organism, and the sample ...
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Estimating Right Censored Data

I'm a newbie at Math, and I am just getting my feet wet when it comes to understanding math and statistics so please forgive my ignorance or question if it's already been answered or if materials are ...
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VST of Poisson random variable of small lambda

https://math.stackexchange.com/questions/805550/variance-stabilization-for-poisson-data As you can see the variance is diminished when the lambda parameter closes to zero. Is there a variance ...
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Proportional Poisson Process Problem

The alliteration is unintentional, but I was amused so I'll leave the title as is. Anyways, I am looking at a problem where we are trying to predict/justify an observed instantaneous rate by which ...
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Modelling poisson distribution with nls() in R?

I'm trying to model the probability that somebody can do a 90 day Snapchat streak. I have some binary success data in the form of count data (the participant only reached 90 days once): I've ...
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