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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Mean of a Poisson-Lognormal Distribution (PLN)

I would like to calculate the mean value of a PLN distribution, $f(x;\mu,\sigma)=\frac{1}{x!\sigma\sqrt{2\pi}}\int_{0}^{\infty}\lambda_\ast^{x-1} e^{-\lambda_\ast} e^{\frac{(log(\lambda_\ast-\mu)^2}{...
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Extension of Poisson Parameter for Different Temporal Interval

Suppose $Y$ follows a Poisson distribution with parameter $ \lambda $ that explains a temporal Poisson process over an interval of 30 seconds. Now, it stands to reason that for an interval of 60 ...
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Pseudo T-stat in Winbugs?

I am trying to obtain the "Pseudo T-stat" in Winbugs for a Poisson Log-normal model. Any suggestion of how can I get that.
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Test and CI for unstable Poisson process

I have empirical data on a process that I assume is Poisson with a given mean, say $\mu$ (unknown). The data is of the form $(x_i, 1\leq i\leq n)$ for $n$ consecutive time periods. I am concerned ...
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Winbugs code multivariate hierarchical Poisson Log-normal CAR model

I am looking for some example code to develop multivariate hierarchical Poisson Log-normal CAR model using Winbugs. Can anyone help me with similar reserach that added their code? Also, how can I ...
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Error in the derivative of Poisson Distribution [closed]

I have a Poisson Distribution $N(x)$ and so I know the error in the $N(x)$. But I want to find the error in $N'(x)$. How to do this?
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How can I decide which element is not following Poisson distribution?

I have a data set which records the items and how many times the particular item is touched. Since each item is independent and the pick is random, we would expect to see it follow the Poisson ...
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Poisson distribution and time intervals

Why is poisson distribution always studied as time interval based when it is just a special case of binomial distribution? Say I have a machine producing pins. X= perfect pin produced (success event)...
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Conditional Expectation (Poisson) UMVUE

Suppose $X_1,X_2,\ldots,X_n$ is a random sample from a Poisson distribution with mean $λ$. How can I find the conditional expectation $E \left( X_1\times X_2\times X_3 \mid \sum_{i=1}^n X_i= z \right)$...
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How to decide if Central Limit Theorem is applicable on a sum of Poisson variables? [duplicate]

I'm confused about the behaviour to expect from a large sum of independent and identically distributed Poisson variables. We know that a sum of $n$ Poisson variables of identical mean $\lambda$ is a ...
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Correlation coefficient of x and y

If we have $$ X\sim Poisson(\lambda), Y|X = x\sim Binomial(x+1,p) $$ What is the correlation coefficient of X and Y? So I used $$\rho=\frac{Cov(X,Y)}{\sqrt{Var(x)Var(Y)}} = \frac{E[X[E[Y|X]]-E[X]E[...
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Interpret interaction terms in Poisson

I'm using GLM to represent a Poisson model with the log link function. The response variable is a count variable representing the number of deaths. Both temperature and season are explanatory ...
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Confusion about the requirements for poisson distribution

Please refers to the below for requirements for poisson distribution. I'm confused about the 2nd bullet point with the 4th. If an event is random, that means the event cannot be associated with a ...
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Comparing Gaussian and Poisson GLM when applied to count data; “Chi-Squared Error”

I have a fixed set of predictors ($[x_1,x_2,...,x_p]$), which I'm using to fit a GLM for univariate responses ($y_1$, $y_2$,...) of various types. E.g. I fit a GLM for $y_1 \sim [x_1,x_2,\dots,x_p]$, ...
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Can you use nAGQ = 0 with the Poisson distribution?

I am working with a GLM with lots of random variables and Poisson distribution. I get the error 'boundary (singular) fit: see ?isSingular' and so looked up ways around this. I found someone ...
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Extremely large (>10000) value of theta in hurdle model

I am estimating a hurdle model with a binomial (first stage) and truncated poisson distribution (second stage). The results look fine but I have a very large value of theta (greater than 10000). I ...
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Why is the Degrees of Freedom 1 higher when model run with Poisson than Negative Binomial

Because the Negative Binomial distribution would result in there be an extra parameter to estimate the over dispersion, I am confused as to why I am getting a DF of 94 when I run the model with ...
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Poisson Process With Two Different Rates

I am looking to investigate a Poisson Process that has two separate rates; essentially, there is a season of games that each have their own points rate (per minute). Comprehensively, over the season, ...
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Camera trapping and the Poisson distribution

I am currently working with data from camera-trapping, where individuals from multiple species have been taken pictures of when they pass in front of one of the cameras (which are placed on a large ...
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How to treat ordinal average rating as a regression predictor?

I have several variables that are averages of responses to a 5-star scale to rate attributes of a product. I have spent a little while reading about the potential issue of averaging a Likert scale and ...
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Overdispersion under different longitudinal constraints

I have panel data (police districts observed across many months) and I am modeling a count outcome. Fitting a Poisson model (simple pre/post comparison with 6 months of data) results in significantly ...
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Zero-inflated Poisson regression with `unit` and `time` fixed effects (Application in R)

In much need of some assistance. My question concerns the conceptualization of zero-inflated Poisson regression in a two-way fixed-effects settings. I have crime data and my outcome is 'count' ...
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Poisson distribution: why does time between events follow an exponential distribution?

I was reading an article, and came across the following: Purchase count follows a Poisson distribution with rate λ. In other words, the timing of these purchases is somewhat random, but the rate ...
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Should I use Poisson or Binomial distribution to solve this question? [closed]

My question goes as below: A major warehouse facility has 10 semi-trailer loading bays, which can all be used simultaneously. On average only 7 bays are in use. a. What is the probability that all ...
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1answer
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UMVUE of the probability a Poisson R.V is odd?

Problem: Let $X_i \sim Pois(\lambda)$. Find the UMVUE of the probability that $X_1$ is odd. My attempt: I don't think there's any obvious unbiased estimator to use conditioning. So instead I write $...
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Is the zero truncated Poisson Distribution part of the Exponential Family? [duplicate]

This is the density of a truncated Poisson: $$P(X = x \mid X > 0) = \frac{\lambda ^ x e^{- \lambda} }{x ! \left ( 1 - e^{- \lambda} \right )}$$ To show that it's member of the Exponential ...
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Concise overview of prototypical distributions

[This question is mainly a reference request.] I'm searching for a somehow concise and complete table of prototypical distributions that would allow a test person to easily choose which typical ...
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p-value histogram of two-sample poisson tests on simulated data

I have two vectors (a and b), which are each a sample of n = 10000 from the poisson distribution with lambda = 10. ...
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Variance of Poisson distribution larger than mu?

So I made a program to calculate variance of Poisson distributions for different $\mu$ and wanted to assert than variance <= $\mu$, but noticed that for larger numbers the variance exceeded the ...
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Comparing assault rates at one facility with two other facilities

Assaults on staff are fairly uncommon in our facilities with between 70-140 in a year at a given facility. Time, similar to 'person years' in epidemiology, is tracked as 'number of bed days' (i.e., ...
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How to calculate poisson approximation?

I have a p-value definition as follows and I would like to implement it. The values of n1, n2, n3, n4, n5 and N are as follows. n1 = 102 n2 = 95 Calculating λ: ...
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Poisson and Gamma distribution for testing randomness

In genetics I want to test whether InDel (insertion and deletion in DNA) sizes occurs with the same probability. I heard that I should gamma distribution to model it. I found ...
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Probability of a event that happens N times in a range of time given others

I'm running an application for making shipments to any place around the globe. I have a set of rules which are like: A customer makes a shipment... Once a week (ie, 50 shipments/year) At least once ...
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Why does this improper prior = constant?

MacKay has an exercise on using Laplace's method for a Poisson model: $$ p(r \mid \lambda ) = \frac{e^{-\lambda} \lambda^r}{r!}, \qquad p(\lambda) = \frac{1}{\lambda} $$ And he asks the reader to ...
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Finding confidence interval for a Poisson parameter that varies over time

I have data as $t=[0, 1 ,2 ,3,4,5,6]$ and $X(t)=[395000,500000,400000,300000,300000,250000,300000]$ and I am fitting to these data assuming $X(t)$~Poisson$X_0e^{-\theta t}$ to estimate the parameter ...
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Poisson distribution for very large numbers - can i decrease the interval?

I want to set up a alert notification if transactions on our website fall above or below a low probability number as it could be due to an error. Lets say i have mean transactions per day of 50,000; ...
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Questions regarding this derivation of the Poisson Distribution from exponential densities

On page 217 - 218 of the pdf of this book, the author derives the Poisson Distribution using gamma and exponential densities. The author defines $S_n$ to be the sum of a sequence of independent ...
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I have derived the Mean and Variance of a truncated Poisson distribution. Does this show under-, equi-, or overdispersion?

The density looks like this: $P(Y=y) = \frac{e^{-\lambda} \lambda^y}{y!(1-e^{-\lambda})}$. I derived the mean and variance and got this: $$\operatorname E(Y) = \frac{\lambda}{1-e^{-\lambda}}$$ $$ \...
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What is the zero-truncated Poisson distribution used for? And how is the mean and variance derived?

I know that the density looks like this: $P(Y=y) = \frac{e^{-\lambda} \lambda^y}{y!(1-e^{-\lambda})}$ and from wikipedia that the mean and variance like this: $$\operatorname E(Y) = \frac{\lambda}{1-...
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Designing hypothesis test around failure rates

I'm trying to devise a hypothesis test for failure rate data of machines. The gist is that there are some machines in a factory that run all the time. They fail from time to time and are promptly ...
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1answer
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Approximate distribution for sum of squares of standardized Poisson random variables

Suppose that $X_1, ..., X_n$ are independent and identically distributed Poisson($\lambda$) random variables. What is a good approximating distribution for $\sum_{i = 1}^{200} \frac{(X_i - \lambda)^2}...
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How to find the “variance” when using “central limit theorem” on a Poisson distribution? [closed]

Assume we have N number of inventors in a company. Inventor i expects to invent X_i number of inventions per year. How many inventions each of them invent per year has a "Poisson distribution" where ...
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Censored regression with Poisson distribution

I am trying to fit a Poisson distribution for left censored data. Let $x_1,x_2,...x_n$ be the observations with the first $r$ observations being less than the threshold of $c$ and hence censored. The ...
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Unbiased estimator for $e^\lambda$ in Poisson distribution

How to generate an unbiased estimator for $e^{-\lambda}$ in Poisson distribution: $\frac{\lambda^k}{k!}{e^{-\lambda}}$ I tried: $$E[a^x]=\sum_{x=0}^\infty a^x\frac{1}{e^{\lambda}}\frac{\lambda^x}{x!}=...
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Finding the MLE of Poisson in R [closed]

I'm trying to determine the MLE of $\lambda$ in a Poisson distribution using R. I'm aware that the MLE is $\hat{\lambda}=\bar{x}$ but I want to demonstrate this using Rmarkdown. My experience with R ...
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Test for difference from 0 for non-normal distribution

I'm using an agent based model to study disease transmission. After the initial outbreak, percentage of population infected decreases to zero. I'm looking for a rigorous way to choose when the % ...
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Order Statistics of Poisson Distribution

I have been given the following question, Let $n ≥ 2$, and $X_1, X_2, . . . ,X_n$ be independent and identically distributed $Poisson (λ)$ random variables for some $λ > 0$. Let $X_{(1)} ≤ ...
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How to measure trends in the frequency of events

for my master thesis i want to figure out which website visitors increased / decreased the intervals between their visits during 3 months. Eventually, I want to segment them into increased and ...
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Distribution of random variable with multinomial sampling distribution and parameters $(n,p)$, where $n\sim$ Poisson with truncation

Suppose you have: $$X\mid N\sim\text{MN}(N,p_1,p_2,\ldots,p_{J})$$ $$N\sim \text{Poisson}(\lambda)$$ What is the marginal distribution of $X$? In this case, the answer is simply this. But... ...
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counting experiment, setting a confidence interval in an input variable

I want to measure the distance between a photosensor and an isotropic light source. The sensor gives me a digital number $x$ as a function of detected light intensity. The light intensity at the ...