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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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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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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 ...
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Bayesian inference on binarized Poisson distribution

I have a variable that is Poisson distributed. Let's say I have a number of boxes each with a number of balls inside according to a Poisson distribution, with $\lambda=0.4$, (the average number of ...
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How to properly truncate exponential distribution to represent random memory-less arrivals?

Typically, Poisson and exponential distributions are used to represent random memory-less arrival processes. It has come to my attention, however, that a more realistic distribution is a truncated (or ...
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What is the distribution of a sum of binomial distributions with the same parameter q but with the sample sizes following a Poisson distribution?

Let $\{a_1,a_2,\ldots,,a_n\}$ be a random sample of a Poisson distribution. Consider the following random variables $X_1=\mathrm{Binomial}(a_1,q), ~X_2=\mathrm{Binomial}(a_2,q),\ldots,~X_n=\mathrm{...
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Finding the mean and variance of an infinite server queue

I am presented with the following homework problem: Let $X(t)$, $t > 0$, be the infinite server queue and suppose that initially there are $x$ customers present. Compute the mean and variance of $...
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Is there a single discrete distribution that handles over and under dispersion? [duplicate]

I have some count data I am trying to model. The variance is very close to the mean, so the Poisson distribution for the entire data set seems like a good starting point. I have done and it seems to ...
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Determining confidence interval with one observation (for Poisson distribution)

When you make only one measurement, how do you determine 95% confidence interval for the Poisson distribution? Let's say that I make one measurement of a quantity $x$ and obtain $x=10$. Also it is ...
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Generating Arrival Counts Given Time

I'm looking to generate arrival counts given a length of time for a Poisson process. This is similar to another question, however rather than generate the particular arrivals, I'd like to just get the ...
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What is Poisson density multiplied by integer-Guassian?

What is Poisson density multiplied by integer-Guassian? $$p(X=x)\propto\frac{\theta^x}{x!}\exp(-x^2),\quad x\in\mathbb{N}$$ Is it a distribution being studied?
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How will p-values behave when fitting normal/poisson to binomial?

I know p-values behave uniformly. Now as p(np) is fixed and n goes to infinity, binomial converges to normal(poisson). Now suppose I take random binomial samplings and fir normal(poisson) to it, for ...
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How can I do a hypothesis test with an estimated null hypothesis?

The situation I have is with data that (I'm assuming for now) follows a poisson distribution. I want to know if this year had an unusual number of a certain type of fatality compared to previous years....
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Poisson process - measuring empirical changes

I was recently discussing Poisson processes in industrial settings with a colleague and he came up with a great question. Let's say that defective products on a manufacturing line tend to occur ...
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GLMMs for count data with glmmTMB: random slopes specification, cross-level-interaction and strange results

folks, I recently found the great glmmTMB package which I hoped would help me with my models. My data are 60,000 facebook posts that are nested in 51 companies (i.e., the posts by these companies). ...
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compute sample probabilities given a poisson distribution

I have read through a large number of stack posts about hypothesis testing on Poisson distributions. Some examples: Calculating test statistic of a poisson distribution Hypothesis testing with ...
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Fitting pmf of a scaled Poisson distribution and Python histogram plotting

I have a nuclei meanlife of $550\mu s$, for which I've taken the frequency(rate) to be $1/meanlife = 1818$. I then sampled randomly from a poisson distribution with that frequency, taking the ...
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Why is this estimator biased?

$X_{1},X_{2},..,X_{n}$ are iid $\sim Poisson(\mu)$ than the MLE for $\theta=e^{-\mu}$ is $\hat \theta =e^{-\bar x}$ Why is this considered to be biased for $\theta$? Is $E[\hat \theta]$ not $\theta$...
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Passion Distribution [closed]

Suppose the number of tsunami in a season follows a Poisson distribution and the average number of tsunami that hit a region is 5 in every tsunami season A tsunami season lasts for 3 months. In the ...
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Using ppois in r to calculate probabilities of random variable with variance

I'm working on a homework assignment on little sleep and hoping for some help. Here is the question I'm trying to write a script for: Write a script in R to compute the following probabilities of a ...
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Poisson regression with continuous data [duplicate]

I have a time series of floating numbers, say, 0.1, 0.5, 1.1, 0.6, 2.0, 1.4, 0.4 Now, I would like to model this series with Poisson regression, since the numbers, even though they seem ...
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Getting 2 defectives

An item is produced by a machine in large numbers. The machine is known to produce 5% defectives. A quality control engineer is testing the items randomly. What is the probability that at least 5 ...
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Expectation on estimator for Poisson distribution

I'm reading through the textbook "All of Statistics" and one of the problems gives the following estimator for the lambda parameter of the Poisson distribution: $\hat{\lambda} = \frac{\sum_{i=1}^n ...
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Which kind of diagnostic plots for count data? [duplicate]

I know that for an lm model is enough to run plot(model_lm) to get diagnostic plots. I am dealing with high-dimensional count ...
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Exponential distribution and Poisson process [duplicate]

Could someone please explain to me what is exponential distribution and poisson process mean? How they are different and the relationship between them? [In simple terms]. Thanks
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Wald test for Poisson distributions

This question asks: $N_A$ and $N_B$ are variables of the counts of the number of events 'A' and events 'B' respectively. Those variables follow Poisson distributions with parameters $\lambda_A$...
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validating probabilistic bug fixes

my apologies if this is too simple to be interesting... SO if I have a program with a problem that occurs at some rate $R$, how many trials do I need in order to be 95% confident that it has been ...
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Poisson vs Gaussian GLM: which to use?

I'm going to provide a simulated case. However, the question is of a general nature (see end of the post). Let's suppose we have some data generated in this way: ...