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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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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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Using binomial distribution in arrivals [on hold]

I have the followig question: Suppose the number of customers arriving is recorded over a 7 days period. Assume that number of customers arriving in a given day are independent. Let Y the number of ...
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Assume that λ is gamma distributed, and y|λ ~Poisson(λ) Derive the distribution of λ given Y [on hold]

Assume that λ is gamma distributed i.e. $\lambda \sim \Gamma(k,\theta ) \hspace{.5cm} 0< k, 0< \theta$ for a given $\lambda$ $Y|\lambda \sim Poisson(\lambda)$ derive the distribution of $\...
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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: ...
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How to model count data with decay

I'm trying to understand how I might model count data where there's diminishing marginal utility and a stochastic process. So, let's say we're modeling the number of "useful intelligence tips" given ...
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Truncated Gamma Distribution

The Gamma distribution is the conjugate prior of Poisson distribution. What about the Truncated Gamma distribution? Is it still the conjugate prior of Poisson distribution?
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Events with two sets of weights - correlated weighted Poisson distributions?

Let's say I have a set of $N$ events with weights $w_i$. $w_i$ follow some distribution, the same for all $i$, that I either know or can approximate. $w_i$ and $w_j$ are uncorrelated for $i\ne j$. ...
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What is the variance of the least of several series?

Given a number of series (or images) that are independent and have a Poisson distribution with the same mean, what is the variance of the series generated from taking the point-by-point minimum? i.e. ...
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Calculating confidence intervals for “excess events”

I have a study that shows 100 deaths occurred in a cohort of high-risk people. Given mortality rates in the general population, you would only expect 70 deaths to occur. The standardised mortality ...
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Estimating population concentrations in spatially autocorrelated data

I'm stuck on which statistic to use with a spatial data set to resolve population concentrations in a large area, when I have only sampled a small area relative to that large area. Here's an example ...
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MLE for Overdispersed Poisson

I searched for a while on Google and this website for an answer to this question. I have an overdispersed Poisson distribution and a "hand-wavy" proof is giving me problems. Below is the information ...
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Negative Binomial Substitute for Poisson Applied to NYC Crime Data

I'm reading these questions and answers (http://study.sagepub.com/sites/default/files/chapter4.pdf) and am confused about 4.2.4 - 4.2.6 I agree that the Poisson model developed earlier is not a good ...
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Mixture of Poissons for positive and negative integers

I'm trying to design a generative model for a random variable $k\in\mathbb{Z}$. The model would work as follows. First draw $b\sim\text{Bernoulli}(p)$. If $b=1$, draw $k\sim\text{Poisson}(c)$. ...
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Distribution/analysis method for small dataset with many small/zero values

I have a relatively small dataset (160 observations), of which a very large number of values for response variables are zero or very small (e.g., 114/160 values are 0; range 0-4250, with only 11 ...
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Fitting the Poisson distribution to binary data

I'm studying several time series. The variables of interests are dummy variables which take value 1 when a certain event happened, 0 otherwise. I want to find the best distribution to describe these ...
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How to handle uncertain counts in poisson test

I am curious about performing poisson test where I have uncertainty about my count. For example, I expect to see 15 bunny rabbits per hike. On a given hike, I positively identify 19 bunny rabbits, and ...
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Can Count data be used as predictors in a linear model

If someone is trying to predict a continuous measure (satisfaction) and their 2 predictors are discrete count data, can that person use a basic linear model (like lm...
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Modeling count data to poisson distribution when expected is zero

I am working on a project where we saw an event occur in 5 of 39 patients. We don't expect this event to occur at all (though we know they occur very rarely, we don't expect to see any in this ...
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Poisson Gamma Distribution in R - Creating Enrollment Modeling Curve

I'm trying to create an enrollment curve for a clinical trial based on the following variables: Country start up timelines (staggered), Number of sites, Number of total subjects needed, Number of ...
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Calculate incidence rate per person or in total

For my study I am doubting about the following: First I want to present the incidence rate for patients who received an CT-scan. I thought I would just count all the CT-scans that took place and ...
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7 accidents per 30 days on average, but variance is 12. Why Poisson distribution can't be used here?

Let's say a city has 7 accidents per 30 days on average. We can use Poisson distribution formula where $\lambda = 7$ $$\frac 1 {n!} (\lambda t)^n e^{-\lambda t}$$ But in Poisson distribution the ...
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Summation Bounds When Finding Transformation of 2 Poisson Random Variables

I am reviewing some material on functions of several random variables from Section 7.4 of John E. Freund's Mathematical Statistics, 6th Edition, and I'm stumped on how the author gets the upper bound ...
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Calculating variance of poisson distributed random variable

I am calculating variance of a Poisson distributed random variable with mean $\lambda$. I am doing it in the following way: $\mathbb{V}(X) = \mathbb{E}(X^2) - \lambda^2 \\ = \sum_{x\geq 0} \quad x^2\...
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two independent Poisson Arrivals

I have two types of customers (type 1 and type 2) enter a shop. Their arrival processes are independent and follow Poisson process with the arrival rates of $\lambda_1$ and $\lambda_2.$ Consider two ...
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Which statistical methods are best suited for distribution with two peaks?

My data shows this distribution: I am looking for a statistical distribution which my data follows. Thought about poisson distribution, but goodness of fit test shows p < 0.05
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Bayes Factor Poisson-Hidden Markov Model

I am following the Hidden Markov Models guide text for Time Series An Introduction Using R (Walter Zucchini). Chapter 7. Bayesian inference for Poisson-hidden Markov models. Specifically in section 7....
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Poisson distribution and completeness, what happens when one point removed from parameter space?

Long time ago (early 1980's) my professor showed me a paper (I think it was Teachers' Corner or something similar) about the Poisson distribution and completeness. Showed that if only one point was ...
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How to calculate the expected value of a function of a Poisson variable

Let $Y \sim Poisson(\lambda)$, and $f(Y)$ is a function of $Y$. Is there a general method, either analytically or numerically, for calculating the expected value of $f(Y)$? In other words, I would ...
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Distribution of a Random Sum [duplicate]

The following experiment is performed: An observation is made of a Poisson random variable $N$ with parameter $ \lambda $. Then $N$ independent Bernoulli trials are performed, each with probability $p$...
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Estimating the mean and the error of estimator for Poisson random variables

$Y_1,...,Y_n$ ~ $P(\lambda)$, $V(Y_i)=\lambda$ $E[\bar Y] = \lambda$, $V[\bar Y]=\lambda/n$ Question is how would I employ $Y_1,...,Y_n$ to estimate $\lambda$, and how would I estimate the standard ...
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Expected time to wait for no events to occur within a sliding window assuming Poissson process

I wish to model the following: I am maintaining a sliding window (history) of 10 samples of the output of a signal detector. I model the probability of a detection failure (i.e absence of signal) as ...
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Convergence of poisson distribution

Let $X\sim \operatorname{Pois}(\lambda)$ and $x_1,\ldots,x_n$ observations following this distribution. I want to derive the analytical solution of the following series: $$\ell(\lambda):=\lim_{x\...
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how to incorporate individual-level classification uncertainty into population-level uncertainty (via Poisson distribution)?

How do I incorporate uncertainty in an individual-level predictive model into the prediction of a population-level variable? To give a specific example: Suppose I have a binary classification model ...
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the power and sample size of one Poisson mean test

I was using the softeware PASS to calculate the sample size and power of a one Poisson mean problem. Solve For ................................................ Power Ha (Alternative Hypothesis).........
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Sufficient statistic for Poisson in wiki?

In Wikipedia: https://en.wikipedia.org/wiki/Sufficient_statistic#Poisson_distribution it says that $X_1+\cdots+X_n$ is a sufficient statistic for the parameter of the Poisson distribution and its ...
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Conditional Probability and Expectation for Poisson Process

To solve part (a) I have $P(X_2 = k\mid X_1 = 1)= \dfrac{P(X_2 = k \cap X_1 = 1)}{P(X_1 = 1)} = \dfrac{e^{-2}}{e^{-1}}=e^{-1}$. Then for part (b), for simplicity, I let $X_2=X$ and $X_1=Y$, then $$E(...
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Calculating Poisson CDF

I have a very dumb question about Poisson CDF. I have the following function $P(x;\mu)$ referring to the Poisson CDF. Hadley and Whitin (1963) defined the Poisson density as $p(x;\mu)= \frac{\mu^x}...
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Conditional probability of Negative Binomial R.V. given the SUM of its values

Suppose $\{z_{ij}\}$ are independent Negative Binomial random variables with means $\{\mu_{ij}\}$, with $i=1\dots I$ and $j=1\dots J$. How do you find the (expectation of) conditional probability ...
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Why doesn't the zero model in a hurdle model exactly match logit result?

I am relatively new to R and I suspect that there is user error here, but I cannot figure out why the output from the logit in the hurdle model does not match the prediction of the "zero" function in ...
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Explanation for Cumulative Distributive Function example

I'd like to ask for clarification of the following example in my textbook. Example: Suppose events are occurring at random with average rate $\lambda$ per unit of time. What is the probability ...
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1answer
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Probability Density function of Poisson distribution

This is an assignment I got for my course on Stochastic Processes: Let us consider a random variable X distributed as a Poisson P (λ) where λ ∼ [0.5, 1]. (a) Which are the unconditional ...
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fitting COM-Poisson in R

I have some crash data I did Poisson for that and the data was underdispersed. I want to do COM-Poisson regression for my data. I see that every website suggest several packages for COM-Poisson and I'...
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How to read checkresiduals graphics in R?

I need to check the residuals of two models in R so I can determine how bad or good are said models. First, I've started simulating an INAR(2) model and wanted to fit a more convenient model, then, ...
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Lower bound critical regions of a poisson distribution

Let's say $P(X=0)$ for a Poisson distribution is $0.18,$ and the $P(X ≥ 5) = 0.04.$ [See OP clarification in Comment.] Obviously, $X ≥ 5$ is a critical region. But because $X ≥ 0$ for a Poisson ...
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Specifying frequency parameter in the absence of occurrences

Let's say I have a process where the occurrences are independent, proportional to time. I made $n$ observations for which I only observed no occurrences. My goal is to define a frequency parameter and ...