# Questions tagged [poisson-binomial]

A discrete probability distribution corresponding to the sum of independent Bernoulli trials that are not necessarily identically distributed.

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### Tight error bound for Possion sampling to estimate a real-valued mean without knowledge of variance

I am trying to develop a tight bound on the sampling error of a Poisson sampling (sampling with non-uniform probabilities). I have a set of real-valued data $X$ with finite size $N$ and I want to ...
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### Extreme birthday problem

I have an extreme version of the birthday problem. I want to know: The probability that $m$ individuals will share a birthday The expected $m$ given the number of individuals The slight complication ...
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### Sum of non-identical Bernoulli is overdispersed or underdispersed Binomial?

Extra-binomial variation is defined in this Oxford Reference source: Greater variability in repeat estimates of a population proportion than would be expected if the population had a binomial ...
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### How can I compare two zero inflated continuous datasets?

I have two zero-inflated datasets such as, dt1= 0, 0.1, 0.125, 0, 0, 1.25... dt2= 1.01, 0, 0, 0.25, 0,... I want to check the differences, like t.test for ...
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### Sum of Bernoulli variables with known probabilities

Following the ideas from this post and, especially, this post, i was wondering if the a sum of two independent groups of Bernoulli distributed variables whose probabilities are know a priori is a ...
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### Can the Poisson-Binomial distribution be used in a generalized linear modeling framework?

I recently came across the Poisson-Binomial distribution while doing some research for a modeling problem I have. I am trying to model the spread of an invasive species where I have county level ...
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### Lower bounds for Poisson Binomial distribution tail probability

Consider a Poisson Binomial distribution $X = X_1+X_2+...X_n$, where $X_i$ is $Bernoulli(p_i$). What I am looking is a lower bound on $P(\sum_{i = 1}^{n} X_i \leq k)$, where $k<\sum_{i=1}^{n}p_i$. ...
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### Hypothesis testing two weighted Poisson Binomial Distributions with different lengths

I have two groups of people: each group is made by subgroups of people coming from census areas of which I know the probability of being male vs female. I can calculate the distribution: is a Poisson ...
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### What is the distribution in Quasi-Poisson regression?

For Poisson regression, the assumption is that Y has a Poisson distribution. Is the same assumption true for Quasi-Poisson regression?
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### How to transform observations between different binomial distributions?

Context I am an instructor and I am trying to grade on a curve -- but using a more rigorous mathematical approach than other quick ways I've seen so far. (I'm also interested in this purely ...
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### 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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### 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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### Poisson distribution and time intervals

Why is poisson distribution always studied as time interval based when it is just a special case of binomial distribution? Say I have a machine producing pins. X= perfect pin produced (success event)...
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### Poisson-binomial vs. Beta-binomial

I have N distinct bernoulli trials with a distinct probability for each trial given by, P=(p1, p2, ..., pN). I want to know the distribution of the number of successes. Given that I know P, I can ...
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### Partial imputation of missing dates

I'm working with dataframes (one for each of 185 locations) that shows sums of occurrences for each calendar date. There are no 0 values for occurrences in the entire dataset. There are several ...
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### Given an observed sample from a binomial distribution, how do I solve for the probability distribution of possible 'p' values?

Let's say I have an 'unfair' coin, for which I'm interested in estimating the 'heads' likelihood or 'p' value. Knowing nothing about the coin, the distribution of probable 'p' values is a uniform ...
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### Sum of independent binomially distributed variables (with different p's)?

The sum of independent variables each following binomial distributions $B(N_i,p_i)$ is also binomial if all $p_i = p$ are equal (in this case the sum follows $B(\sum_i N_i, p)$. If the $p_i$ are ...
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### How do I propagate error for a Poisson-binomial distribution (sum of probability estimates with standard deviations)

My question is about quantifying the uncertainty associated with a sum of several probabilities, in a case where the probabilities are unequal and are themselves estimates with associated uncertainty (...
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### Probability mass function of poisson binomial distribution

I'm trying to write a program that calculates the PMF of a Poisson Binomial Distribution. I know there are libraries that have all sort of probability distribution functions, but I'm doing this just ...
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### Probability mass function with variable probability? [duplicate]

I would like to calculate the probability of a certain result coming out at least 'x' times in 'n' attempts when the probability of result varies on every attempt; all attempts are independent events. ...
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### Comparing two rates: Poisson test or Fisher test (or logistic regression)

I have a setting in which I want to compare the rate of two count quantities under two different conditions. This can be achieved in two ways: either using a Poisson test, comparing the cases and ...
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### MLE for Poisson-binomial distribution

I am looking for the maximum likelihood estimator (MLE) for the Poisson-binomial distribution. I understand the derivation of the MLE for a Poisson distribution and a binomial distribution, but I am ...
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### How to go from given incidences in 24 years to annual probability?

For a biostatistics class, I have to determine the annual probability of at least one heart attack for various groups of given data that was collected over a period of 24 years. I won't put all the ...
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### Standard error of the success rate of a collection of different binomial distributions (whose success rates need to be estimated)

I would like to estimate the correct percentage of tweets that are written in a particular language from a collection of followers of a given account . The goal is to compare different accounts to ...
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### Poisson Binomial Distribution

What are the best possible bounds for the Poisson Binomial Distribution's tails? I need the bounds, because in my problem, I don't have the exact value of the probabilities $p_i$ related to the ...
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### Probability distribution of two different probabilities

I have a problem that I was able to reduce to the following statement: What is the probability of throwing at least k ones when throwing n fair six-sided dice and m fair eight-sided dice? ...
Assuming $n$ independent Bernoulli trials with different probabilities, the Poisson binomial distribution is the discrete probability distribution that describes the number of $X$ successes. A ...