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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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1answer
32 views

How can I describe a heavy skewed population?

I have a set of data on the size of a group a person is apart of for 100 persons. member group_size bob 16 sally 30 jim 5 The size of the group a person ...
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1answer
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Poisson distribution question

An airline has found that the number of people booked on flights who do not arrive at the airport follows a Poisson distribution at the rate of 2% per flight.For a flight with 146 seats ,150 are sold ...
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24 views

Poisson distribution - adapting for different intervals

Over a long period of time ,a plumber finds that on average he receives 2 emergency calls per week .Work out the probability of one emergency call on one day assuming the plumber is available for ...
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1answer
24 views

Are these two distributions independent?

In a probability cheatsheet, there's the following claim: That $Z_1$ and $Z_2$ should be independent seems very unintuitive to me. It seems to me that if $Z_1$ is high, then $Z_2$ is low, and vice-...
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1answer
34 views

Poisson distribution with large mean

The hits in a website can be modeled as a Poisson process with a rate of 5000 hits per day. Let X be the number of hits in 6 days. Give the distribution of X. So, is the percentile x = 6 and the ...
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53 views

Confidence interval over multiple Poisson distributions

There are five different periods (one following each other) of unequal lengths where one specific event is happening such that it follows Poisson distributions (e.g. meteorite falls). It is always the ...
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1answer
19 views

Adding probabilities of different Poisson distributions

There are three different periods of unequal lengths where one specific event is happening such that it follows Poisson distributions (e.g. meteorite falls): in the 1st period (100 hours) the average ...
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1answer
28 views

Conditional independence problem for poisson random variables

I have this problem: Let $X = V + W$ and $Y = V + Z$ where $V, W, Z$ are independent Pois($\lambda$) random variables. I found that $Cov(X, Y) = Var(V) = \lambda$ It now asks to find whether $X$ ...
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34 views

How can I apply the Poisson ($\mu$) distribution to two series of random draws?

I have the following question: A box contains 1000 balls, of which 2 are black and the rest are white. If two series of 1000 draws are made at random from this box, what approximately, is the chance ...
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1answer
22 views

Finding joint distribution of sum of independent Poisson random variables and calculating probabilities [duplicate]

My textbook, Introduction to Probability, by Blitzstein and Hwang, provides the following problem: Let $X = V + W$ and $Y = V + Z$ where $V$, $W$, and $Z$ are independent $\text{Pois}(\lambda)$ ...
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1answer
360 views

Probability of a 500 year flood occuring in the next 100 years - comparison of approaches

I'm looking at this problem A $500$-year flood is one that occurs once in every $500$ years. a) What is the probability of having at least $3$ such floods in $500$ years? b) What is the ...
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DIspersion Parameter of Generalized Poisson Distribution

Could someone please explain how the dispersion parameter in a generalized Poisson distribution is calculated? I'm currently trying to write an EM algorithm for a zero-inflated generalized mixed ...
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Estimate quasi AIC for GAM models in R

I am new to R with little statistical backround. I want to built a GAM model which consists of categorical and continuous variables and cross basis matrices(cb). I use dlnm and mgcv packages. So, I ...
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1answer
35 views

Poisson regression with offset vs poisson regression with weight [duplicate]

I would like to know the linear expression of weight and offset in terms of poisson regression in glm. for instance for offset ...
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3answers
79 views

Standard deviation of a scaled Poisson process

Assume we have some Poisson process that produces events. In a given year we have counted $N$ of these events. Further assume that for some reason we need to report a monthly rate instead of this ...
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15 views

Using the Neyman-Pearson lemma to construct UMP test for Poisson rates

We have two Poisson processes with rates $\lambda_1$ and $\lambda_2$. We observe $n_1$ events in $t_1$ time from the first process and $n_2$ events in $t_2$ time from the second. The UMP test for ...
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If $X~Poisson(\lambda)$ then does $2X$ also follow Poisson with parameter $2\lambda$

I have a confusion in the parameter part. I know that the sum of n i.i.d Poisson variates is also a Poisson variate with parameter as the sum of individual parameters. For instance, if $X_i$~$Poisson(...
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35 views

Determining Link for GLM

I am having a great deal of difficulty understanding how to use the Generalized linear model for my data set. The response variable of interest is hatch success of sea turtles, which is a %. The ...
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Simultaneous Poisson processes

The arrival of taxis at a taxi stand is Poisson at rate $\lambda$ per hour. The arrival of people to the stand is also Poisson but at rate $\mu$ per hour. Taxis do not stop (they leave empty) if ...
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1answer
29 views

How to approximate the distribution of the sum of multiple multinomial random variables?

Say we have $T$ independent Multinomial random variables $X_1,X_2\dots X_T$, with $p(X_t=m)=p_{t,m},m\in\{1,2,...M\}$. What would be the distribution of $X_1+X_2+...+X_T$? If there is no closed-form, ...
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1answer
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One- vs two-sided credible interval for Poisson process with all zero counts

In a related question, I asked about a confidence interval for the estimate of the mean of 50 observations of a Poisson random variable, for which all 50 observations had a count of zero. In the ...
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1answer
62 views

How to find an expression of the variance of a Poisson-Lognormal distribution?

I am using a model for the number of goods in a supermarket cart with a Poisson-lognormal distribution (a lognormal mixture of Poissons). I would like to find an expression of the variance of this ...
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1answer
41 views

Question for finding Probability for sum of two Poisson Distribution

A survey of downtown Chicago found that the typical city block had an average of five rats per block. This was true irrespective of whether or not there was construction occurring on that block. If ...
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1answer
72 views

Probability of multiple parasites given the presence of one parasite

I am trying to simulate parasitic plants that infect trees. I would like to test wether the presence of one parasite makes it more likely that the tree has more than one parasite. The number of trees ...
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2answers
120 views

Confidence interval for mean of Poisson with only zero counts

I have 50 sample units in group A all with counts of zero. I have other groups that have some non-zero counts. I would like to make a 95% confidence interval for the mean count per sample unit within ...
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Sum of many poisson random samples gives approximately normal distribution. But my events are still rare [closed]

It is well know that if $X_1,X_2,.., X_n$ are all independent and sampled from $Poisson(\lambda)$, then $\sum_{i=1}^n{X_i}\sim Poisson(n\lambda)$. I have a following situation which is similar to the ...
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Find significant neighborhoods with increase or decrease in #CurrentVisits compared to #BeforeVisits

I am currently researching on some patients data to understand the effect of a hospital's inauguration over the current visits of the patients compared to their Before visits. The study area contains ...
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2answers
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Hypothesis test for Poisson process failure rates stays just as powerful for Compound Poisson

The book "Testing Statistical Hypotheses" by Lehman describes in section 4.5, a test for comparing the rates of two Poisson point-processes. This is the "uniformly most powerful" (UMP) test for this ...
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Does it make sense to infer a rate (as a probability distribution or upper limits) for a Poisson process if there are “no events”

I have an inhomogeneous Poisson process with a rate $\lambda (\mathbf{t})$ defined on some parameters $\mathbf{t}$. I am trying to infer $\lambda (\mathbf{t})$ from some data, which are events (really ...
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1answer
42 views

Weighting data based on the errors

I have some data (counts) with a Poisson error associated with them and I want to fit the data. I am trying to weight the data inversely proportional to the errors, such that the data points with high ...
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1answer
42 views

Passenger Inter-arrival times

A number (n) Cruise ship passengers disembark at a port of call. n people will arrive on 5400 second interval prior to the ship's departure. then the average inter-arrival time is 5400/n = 54 seconds. ...
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1answer
76 views

Confidence intervals for the mean of a sample of counts

I'm sure the answer to this question is obvious, but I can't seem to find it! How do you calculate the confidence interval around the mean of a sample of counts? I've made an example below in R, ...
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Derivation for mixed distribution, Poisson-Lindley

I want to derive the Poisson Lindley Distribution. $$ f_x(x|\lambda) = \frac{\lambda^{x-1}}{(x-1)!}e^{-\lambda}$$ $$f_x(x|p) = \frac{p^2}{(p+1)}(\lambda+1)e^{-\lambda p}$$ The Distribution of x, $...
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Test design Poisson Rates

I would like to design a test for the following problem, where the Poisson distribution applies: Samples from a population can be either defective or not (e.g. some technical component from a ...
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1answer
26 views

Poisson regression for count data that is not Poisson distributed?

Is it true that Poisson regression is used to model count data? But not all count data follow a Poisson distribution? Then you can still use Poisson regression in that case?
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Comparison of Poisson cumulative probabilities between models with different rates

I'm building an anomaly detection model using the Poisson distribution. I'm calculating the Cumulative Probability: P(X >= x) for thousands of count data, all of them with different rates. I need to ...
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How to generate under-dispersed Poisson variables in R?

In this article "generate quasi-poisson distribution random variables", it is shown how to use the function rnbinom to simulate overdispersed Poisson data: ...
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1answer
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Creating a classification model for a target value with Poisson distribution

I have a dataset with a target variable following Poisson distribution. It starts at 0 and goes until 30. But there are 8300 0's and only 2 30's. From this data, I need to create a classification ...
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2answers
109 views

Possible methods for parameter estimation of a compound Poisson

Let $X_i$ be iid and each $X_i$ takes one value among $(0,1,-1)$ with probs $(p_1,p_2,1-p_1-p_2)$ respectively. Let $N$ be a Poisson RV with mean $\lambda$, and $$Z=\sum_{i=0}^N X_i$$ be a compound ...
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Is there a difference between a “binary Poisson variable” and a “Bernoulli random variable”?

I am currently reading this paper which includes some stochastic modeling, and the authors describe a term in their model as being a "binary Poisson variable" which takes the value 1 with (time-...
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Modeling variable recensus/exposure count data using gam, quasipoisson with cloglog link

I am modeling tree recruitment in a forest. Foresters visit a forest monitoring plot twice. Recruitment is defined as the number of trees above a certain size threshold that are in the second census, ...
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1answer
52 views

How to use poisson to estimate arrival time (generate random integers)?

In my assignment, I'm supposed to use a distribution curve to describe how many minutes will the employee come to work in advanced or later than the planed start time. I decided to use Poisson ...
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1answer
87 views

How to write the log likelihood for the sum of independent poisson events

Say that the per each event of TYPE 1, the average number of occurrences is $\lambda_1$. Then the likelihood for the number of occurrences in a single event, $k_1$, is $\lambda_1^k / k_1! * e^{-\...
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How does R's “poisson.test” function work, mathematically?

I am trying to understand the mathematics underneath the poisson.test function in the stats package in ...
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1answer
41 views

Data set with a distribution similar to Poisson, which regression to use?

I am working on a research model looking at the data from stack overflow, especially the relationship of various variables on the score of a question. looking at the histogram of my dependent variable ...
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1answer
46 views

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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1answer
25 views

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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1answer
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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 ...