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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Poisson process and uniform distribution

Question: A single-pump petrol station is running low on petrol. The total volume of petrol remaining for sale is $100$ litres. Suppose cars arrive to the station according to a Poisson ...
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36 views

Poisson distribution with exact value of x and mean is a larger number

$X $ follows $ Po(80)$. I used a normal approximation to get $P(55\leq X\leq 75)$ which I got correct. I need to find $P(X=80)$. I tried the Poisson distribution directly; however, my calculator ...
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28 views

Exponential Distribution Expected Value

Question 1 (part 1) I was attempting to use the exponential distribution to solve the following problem: the time between cars passing on a road can be modelled with an avg rate of 3.5 cars per ...
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29 views

Which machine learning method to use for geographic systems prediction?

I am trying to do experiments on geographic systems prediction. We're working on classifying the location where we sell product most. So, we need to analyze the hestorical data and predict the success ...
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1answer
30 views

Continuity correction error when using normal distribution to estimate Poisson distribution

Good morning Everyone, I am doing a self-study exercise which attempts to exemplify a case where the Normal Distribution is used to approximate the Poisson Distribution, since the population mean is ...
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48 views

Offset variable in Poisson regression makes model worse

I'm using a Poisson regression to model count data (number of orders). My observation lengths are different, so I tried to include an offset variable in the model. The problem is that when I estimate ...
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45 views

Poisson distribution confidence interval

Edit : this is the data I used for the first part of the problem : \begin{matrix} Rocks & 0& 1& 2& 3& 4& 5& 6 & 7\\ samples & 12& 27& 28& 19& ...
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31 views

POISSON regression with offset variable

Our problem is to determine if there is a relationship between the return on equity of firms (ROE) and the presence (number of indicators used) (Y) of a specific type of indicator showing up in the ...
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77 views

Understanding factor potentials in PyMC

I'm trying to understand factor potentials from the PyMC documentation, but need some help on the implementation piece--or it may turn out that I am misunderstanding how potentials work altogether. ...
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28 views

How to interpret coefficients in a Poisson regression with interaction terms?

This question is a prolongation of this question: How to interpret coefficients in a Poisson regression? If we follow the (almost) exact same routine, but we add correlation between the variablese ...
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35 views

Sufficiency of two Poisson disributions

If $X_1,X_2$ constitute a random sample of size n=2 from a Poisson Population show that the mean of the sample is a sufficient estimator of the parameter $\lambda$ . Since the sum of Poissons is ...
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34 views

Standard error of a sum of two Poisson variables

I have 2 Poisson variables, I know mean and standard error for each. How do you calculate the Standard error of the sum?
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44 views

Poisson distribution help

In many cities, neighborhood Crime Watch groups are formed in an attempt to reduce the amount of criminal activity. Suppose one neighborhood that has experienced an average of 10 crimes per year ...
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Back Transforming Rates in Poisson GLM with Box and Cox Transformation

Suppose I have fitted a Poisson GLM to model rates as follows: > fit.1=glm(response~X1+X2+ offset(log(population)),family=poisson,data=...) I can get the ...
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94 views

How to use the chi-squared test to determine if data follow the Poisson distribution

The figure below (Figure 1 from p. 646 of this paper) compares observed values against expected values under the Poisson distribution. It then runs a chi-squared test to see if the observed values ...
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136 views

Choosing a discrete non-uniform distribution for generating random integers

I have a list $l$ containing integers in the range $[1,max]$ On list $l$ I do an operation $isPresent(x)$ which return true if ...
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1answer
70 views

Poisson distribution of two independent random variables

Let X1 and X2 be two independent random variables. Let X1 and Y=X1+X2 have Poisson distributions with means μ1 and μ>μ1, respectively. Find the distribution of X2. I don't know how to start solving ...
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61 views

Using Poisson distribution to generate random integers

I'm trying to generate random integers which have Poisson distribution. The open source library GSL has one such distribution. Function: ...
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12 views

Confidence interval for fit to poisson count data, for beginner

I have the following graph, which I then normalise and attempt to fit to. The data is a histogram of counts at a given time. The fit then looks like: The issue is that the parameters ...
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21 views

Error bar for Poisson count data

I have a set of data, counts versus time. The whole data looks like this Here is a sample ...
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43 views

Testing if alcohol consumption and smoking are independent

The question asks to test if smoking status and level of alcohol consumption are independent using the usual five-step procure at alpha $=0.05$: I am having trouble finding expected values. As the ...
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45 views

Total waiting time at a server with exponential arrival times

Question: In a certain system, a customer must first be served by server 1 and then by server 2. The service time at server $i$ is exponential with rate $\mu_i$, $i = 1, 2$. An arrival finding server ...
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24 views

Using residualized predictors outside the linear model context

Can anyone point me towards a good explanation of when a residualized variable in a regression will give you the same answer as using a non-residualized variable with controls? For instance, say I ...
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29 views

Testing a hypothesis of H0: $\lambda>=0.02$ against H1: $\lambda<0.02$

I'm not sure where to even begin this problem. Possibly something with taking the ratio of the likelihoods for the null and alternative?
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474 views

Same or different? The Bayesian way

Say I have the following model: $$\text{Poisson}(\lambda) \sim \begin{cases} \lambda_1 & \text{if } t \lt \tau \\ \lambda_2 & \text{if } t \geq \tau \end{cases} $$ And I ...
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28 views

Laplace rule of succession for a Poisson process

Suppose that I measure for an interval of time $t$ a Poisson process that has unknown average rate $\mu>0$ and I observe no event. What could be a reasonable estimate of $\mu$ from a Bayesian point ...
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2answers
41 views

Changing the units for an independent variable in a Poisson model

I'm trying to model a dose-response relationship between the incidence of cancer (dependent variable, counts) and the radiation dose (independent variable in gray). Can I change the unit of the ...
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42 views

Maximum likelihood of two Poisson distributions

Suppose I have two sets of samples $\{a_i\}$ for $i=1, \ldots, n_a$, and $\{b_j\}$ for $j=1,\ldots, n_b$, drawn respectively from two Poisson random variables $A \sim\textrm{Poiss}(\lambda_a)$ and $B ...
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1answer
34 views

Should anova(model, test=“Chisq”) not produce a test statistic as well as a P value?

Doing a poisson regression like this: model<-glm(y~x*z,family=poisson) with one predictor being a factor, I would use ...
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94 views

Obtaining $\lambda$ for a Poisson Distribution

Is it possible to manipulate a Poisson Distribution's $\lambda$ value such that it has a probability mass function that looks like the curve below with data points $(8,0)$,$(7,0.01)$,$(5,0.1)$ and ...
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55 views

Poisson vs Binomial for rare events

From Poisson's postulates, we know Poisson works for rare events. However, we also know binomial is an approximation of Poisson when the probability of an event is small. So can we use binomial and ...
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Convergence in Distribution\CLT

Given that $N = n$, the conditional distr. of $Y$ is $\chi ^2(2n)$. $N$ has marginal distr. of Poisson($\theta$), $\theta$ is a positive constant. Show that, as $\theta \rightarrow \infty$, $\space ...
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37 views

Zeros in non-negative data: can you multiply by 10^c, round, and then model as poisson GLM?

Potentially stupid question: is there anything wrong with modeling zero-containing non-negative data by first multiplying it by 10^something, then rounding it, and then modeling it as poisson? Or is ...
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35 views

Poisson random variable conditioned on sum [duplicate]

If $X_1,X_2...X_N$ are independent Poisson variables with parameters $\lambda_1,\lambda_2...\lambda_3$, then given $\sum_iX_i=N$, we have that $X_i \sim ...
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52 views

How to interpret output of poisson.test?

Chapter 11 of Introduction to Data Science is about Poisson distributions. One example sample has 58638 observations out of 100000 with a value less than or equal to 10. ...
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20 views

Independent vs. Dependent Compound Poisson Distributions

I have an issue with a section of some Actuarial lecture notes that I am reading. Here are the snippets from the notes: "Consider a portfolio consisting of $n$ independent policies. The aggregate ...
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1answer
44 views

Parameter Estimation of a Poisson mixture model

I want to estimate the parameters for a Poisson mixture model with 2 (and later 3) Poisson distributions. I want to use Matlab and have numerical problems to solve the loglikelihood of the mixture ...
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1answer
59 views

Fitting a copula with Poisson marginals to data in R

First off, I know this is a question which requires an thorough answer, so I am coming here with a very humble attitude. I have limited knowledge about both copulas and R, so I will try to explain ...
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1answer
51 views

How to calculate Mid-p confidence intervals for SMR, SIR (observed/expected)

I need help calculating the Mid-P of the confidence limits for the standardized mortality ratio, or any observed/expected ratio. I am familiar with the Mid-P for a binomial calculation but I am trying ...
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59 views

Which is the difference between a distribution and a process (Poisson)?

I'm doing my PhD in geomechanics. I thought we use a Poisson-Weibull distribution (for the variability of a parameter at the rock), but reading more about the subject I think maybe is a ...
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56 views

Proving no unbiased estimator exists for $\theta^{-1}$ for Poisson Dist. Why is $X_1,…,X_n = \sum{X_i}$?

Suppose that $X_1,...,X_n$ is an iid random sample from a Poisson distribution with mean $\theta$. I would like to prove that there exists no unbiased estimator of $\frac{1}{\theta}$. To do so, I ...
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1answer
55 views

Statistical test for # stores per capita - exact Poisson or t-test?

I have the population denominators for high (n = 28137) and low (n = 35167) deprivation areas in a city. I want to be able to compare the # of convenience stores per capita in high and low deprivation ...
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36 views

Poisson regression versus Poisson random effect regression models

i have a fitted a classical Poisson regression model to my 584 claim frequency data set for a period of 5 years (panel data)i have found out that predictors area,age and make of vehicle are ...
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57 views

Steps required to choose a suitable model for Poisson regression

I have data on claim frequency and several predictors like age, gender of insured person and characteristic of vehicle. I have checked for the assumption of mean equal to variance of a Poisson model. ...
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Measuring the performance of a classifier that estimates bin counts for a Poisson process

I have some data that results from the convolution of a Poisson random process with a decaying exponential, plus some additive Gaussian noise. I'm working on designing a classifier that returns a ...
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Exposure in Negative Binomial regression and other distributions

For a Poisson regression we can model exposure $\epsilon_i$ in observation $Y_i$ as $Y_i \sim Poisson(\epsilon_i*\lambda)$. For instance, in a Poisson regression, if we observe: $y = \begin{bmatrix} ...
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1answer
91 views

When to use Poisson distribution?

I'm slightly confused with when I should use a Poisson distribution. If I have a case where: I have the sample mean of the number of events happening in a particular day (say it is 1.27). Five ...
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1answer
105 views

Variance of arrival process with shifted exponential distribution

Here we have a arrival process. The inter-arrival time follows a shifted negative exponential distribution. The density function of the distribution is: $$f(t)=\lambda ...
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83 views

Exponential Distribution, Poisson process

I have a question relating to Poisson which I would like to get some advice on. I am doing this on a self-help exercise, but unfortunately it doesn't provide answers for me to verify. Basically in ...
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50 views

How to calculate the expected zeros in a Poisson distribution?

I am modelling the nights spent at hotels (count data) fitting a few predictors in the model. I'd like to know how to calculate the expected number of zeros in this distribution, as I suspect that I ...