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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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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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11 views

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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1answer
34 views

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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17 views

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
72 views

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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15 views

fitting COM-Poisson in R

I have some crash data I did Poisson for that and the data was underdisperesed. 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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1answer
38 views

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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44 views

Probability that exactly 12 buses will arrive within 3 hours

Let's suppose there are two buses $A$ and $B$. They draw up at the bus stop under the Poisson distribution with intensities $3$ and $5$ times per hour. (a) What's the expected length of time after the ...
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1answer
44 views

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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1answer
40 views

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 ...
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1answer
74 views

regression coefficient in the poisson model [closed]

When we are dealing with count variables we are told not to log transform our data but to instead use a poisson regression. I was wondering.. when it comes Poisson regression, the common formulae is :...
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How can i find the following Poisson distribution when i have a non-integer value [duplicate]

I need to find the Poisson distribution of successes that occur in a 30 minute period based on a 3% per hour success rate, i know the Poisson distribution formula (P.M.F) but can't figure out how to ...
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1answer
43 views

Newsvendor problem with Poisson demand

Problem: The demand for a particular weekly magazine in a newsstand follows a Poisson distribution averaging 5.2 copies per week. The value paid for each magazine is 15.00 and the sale price is 30.00....
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8 views

Statistical significance of bin content of an histogram and the connection with Poisson uncertainty

I'm considering histograms represent scalar functions, $f(x): {\rm I\!R}^n \to {\rm I\!R}$, where at most $n=2$. I read that the content of each $i$th bin, $N_i$, should be statistically significant, ...
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1answer
17 views

Compound risk poisson models

I was just working through this question. A compound Poisson risk model is used to model the total claims S experienced by an insurance company over one year, of the form: $S = X_1 + ... + X_n$ ...
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92 views

$X$ and $Y$ being two independent Poisson random variables

Let $X$ and $Y$ be two independent Poisson random variables, with means $\lambda_1$ and $\lambda_2$, respectively. Then, $X + Y$ is a Poisson random variable with mean $\lambda_1+ \lambda_2$,. Arguing ...
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1answer
61 views

Correcting data using poisson-regression

I'm new to stats and I was wondering if anyone had any good resources that could explain to me: How one can correct their data (false-positives) using Poisson-regression. I've been looking for some ...
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24 views

Poisson adaptation of IRT model: how to interpret parameters?

for my work I made an assessment in which users have to stack blocks in a certain configuration in as few steps as possible. If the user does this in the minimally required number of steps, they get ...
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1answer
39 views

Calculate threshold value for Poisson distributed noise

I need to calculate a threshold value to get rid of Poisson distributed noise in an image to perform a cluster analysis on the image. The image is the representation of a signal, whose datapoints ...
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32 views

Does the independence of samples in an experiment matter?

This is a bit complicated. I have a set of data of events happening in time, so my data looks like an array of time points at which an event happened. (Eg [2 45 50 51 60 79] in seconds) This data was ...
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1answer
71 views

Poisson distribution - Number of accidents

I need help with this probability problem. The number of fatal car accidents that happen in a specific region follows the Poisson distribution with a rate of 0.5 fatal car accidents per day ...
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16 views

Offset in NB model produces contradictory results to original data

My (example) data frame is rather simple. I would like to know if there is a difference in numbers between the factors (n=2) within variable X. ...
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0answers
26 views

Suggesting a method of moments estimator for the chance that some event happens

Let $X_i$~ $\text{Pois}(\lambda)$ be the number of breakdowns a certain ATM machine experiences in the $i^{th}$ week. $\implies$ Let $ \{X_i\}_{i=1}^n$ be iid of the number of breakdowns the machine ...
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Zero-inflated Poisson distribution parameter estimates

Let's say we have a population distributed by Zero-inflated Poisson distribution: $$ f(x | \psi, \lambda) = \left\{ \begin{array}{ll} 1-\psi + \psi e^{-\lambda} & \mbox{if } x = 0 \\ \psi \...
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Identifying the distribution of this data

I'm working with a datetime series, and each observation is the count of occurrences per that period. This sounds to me like these values would be Poisson distributed. But my EDA has left me ...
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How to write the Bayesian classification rule when probabilities are known and unknown?

Consider classification based on Bayesian rule where classes $\Omega_l$ have prior probabilities $\pi_l$ and training samples have Poisson distributions with parameters $\lambda_l$, $1,\dots,L$. Write ...
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1answer
26 views

How to use Poisson Lambda

I am writing a piece of software that requires Poisson distribution with an average, or lambda, of 5 events per seconds. I am passing the lambda in a loop that "sleeps" a number of milliseconds to ...
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Why there are not (long tail) alternatives to dirichlet-multinomial (while there are for posisson-gamma)

While there are a lot of long tail alternatives to poisson-gamma (negative binomial), for example (Source) I haven't found any work on replacing the dirichlet distribution with a more long tailed ...
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1answer
40 views

Writing Likelihood of Poisson in R

Here is my attempt to make the likelihood function for Poisson distribution for data x and parameter theta in R: ...
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0answers
67 views

Expected value of quotient of Poisson distributions

Let $X$ and $Y$ be independent random variables such that $X \sim \text{Poisson}(\lambda \cdot c)$ and $Y \sim \text{Poisson}(\lambda \cdot (1-c))$, where $c$ is a real number in $[0, 1]$. Is there ...
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1answer
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how to find correlation coefficient when X and Y follows Poisson Distribution?

A bridge is examined for corrosion. It is believed that the corrosion on left side exist is poisson distributed with mean 3 and corrosion on right side is poisson distributed with mean 1.5+0.5X where ...
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Determining the type of data distribution

I am looking on expanding my introductory knowledge in stat, and I have come up with this challenge. I have an application that is designed to analyze and index news articles. The goal of the ...
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Long tailed count probability distributions - with available likelihood calculation and random number generation

in real world data, often (always to me) happen that for modelling count data Poisson Negative binomial multinomial dirichlet-multinomial probability distribution, are not robust to outliers, as ...
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1answer
23 views

How should I implement the cumulative distribution function of a discrete r.v that follows binomial distribution in R?

Problem Let there be a discrete random variable s.t. $$X \sim \text{Binom}(20,0.02)$$ and $X(\Omega) = \{0,1,2,\ldots,20\}$ Let there be also a constant $C$ s.t. $$\Pr(X\leq\frac{120}{C}) &...
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1answer
51 views

Upper bound on $P(n^{-1}\sum_{i=1}^n (X_i - \lambda_i)>t)$ for independent $X_i\sim\operatorname{Poisson}(\lambda_i)$

Let $X_1,\dots,X_n$ be independent random variables, $X_i \sim \operatorname{Poisson}(\lambda_i),$ $i=1,\dots,n.$ Let $$S=n^{-1}\sum_{i=1}^n X_i, \quad\quad \lambda=n^{-1}\sum_{i=1}^n \lambda_i.$$ ...
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Normalize Count Data (Poisson Distributed Data) with many Zero Values

I have a count data set for the number of couns of an alert(over speeding alert) for various vehicles(number_rows=206). Please assume this count data is for vehicles who have covered similar amount ...
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24 views

Comparing two sets of exponential data $T(t)=−e^{−kt}$

I have two sets of exponential data (temperature measurements) of the form: $T(t)=−e^{−kt}$. k is a constant that determines the rate of temperature change. 1 temperature measurement was taken every ...
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Does exponential waiting time for an event imply that the event is Poisson-process?

Say I have a process, $\{N_t : t \ge 0\}$, which denotes the number of the event that occurred until the time $t$. And let me define $W = \min \{t : N_t = 1\}$ which is denotes the time until the ...
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33 views

Confused about Poisson Distribution [duplicate]

In textbooks, the Poisson distribution is given as $P_\mu(x)=\frac{e^{-\mu} \mu^x}{x!}$. While in Random Matrix theory, they call this: $P_\mu(x)= \mu e^{-\mu x}$ as Poisson distribution. Maybe ...
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1answer
47 views

Multinomial vs Poisson for Histograms

Preliminaries My question came about when reading Lawrence and Chromy's paper on Maximum Likelihood (ML) estimation of histograms (link). I am aware of this question, but I fail to see how that helps ...
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51 views

Test for equality of means of two paired count variables

Is there any significance test to compare count data measured at pre and post from the same subjects? For example, compare the number of car accidents of a group of drivers at their first year of ...
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0answers
24 views

How to calculate the average number of events per interval in poisson distribution

Hi dear statisticians, I have a random variable X that follows a poisson distribution. However, I only know the number of occurrences in an interval and the cumulative probability a. How I can ...
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1answer
52 views

Does taxi demand follow a poisson distribution?

The important assumptions underlying a Poisson process are: What happens in one subinterval of time is independent of what happens in any other subinterval The probability of an event is the same in ...
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1answer
31 views

Poisson distribution: how to calculate the expected number of zeros in the presence of an exposure term

I have a dataset each with a line for different products, a count of sales and a number of days on sale. I estimated a Poisson regression to predict sales with an offset term of number of days on sale,...
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1answer
233 views

Simulate MLE for Poisson distribution [closed]

According to the theory given $X_i$ ~ $Pois(\lambda)$ iid, the maximum likelihood must be equal to $\sum_{i=1}^{n} X_i/n$ in this case $5.01$ ...
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1answer
103 views

Poisson Sample Variance

$Y_1, ... , Y_n$ are taken from a Poisson distribution. There are two estimators, $\mu_1 = \bar{Y}$ $\mu_2 = Y_1 + Y_2 / 2$. Show with a mathematical proof that $\mathrm{Var}(\mu_1)$ is less than $\...
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49 views

Explain sufficient statistic for Poisson distribution [duplicate]

The Wikipedia entry on this topic is, to me, very confusing. It states that: If X1, ...., Xn are independent and have a Poisson distribution with parameter λ, then the sum T(X) = X1 + ... + Xn is a ...
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19 views

Expectation value of product of kronecker deltas

Consider the sum squared deviation $\text{SS}$ of $N$ samples $\hat{x}_1,\ldots,\hat{x}_N$ of a poisson random variable with mean $\mu$, i.e. $$ \text{SS} = \sum_{m=0}^{\infty} \left( \sum_{n=1}^N \...
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Prediction interval for sampled count data

I am trying to get prediction intervals around a sampled count variable. For example, say I want to know the number of letters an apartment building receives every day. Each day I record the count ...