# Questions tagged [poisson-binomial-distribution]

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

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### Modelling probabilities of a sum of binomials with different probabilities and trials

I have the following example data, where each row is an independent observation: A B C Y 10 22 6 2 4 60 2 0 12 8 10 3 ... $A$, $B$, $C$ and $Y$ are all positive integers. The variables $A$, $B$ ...
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### Finding Correlation between Defect Rate and Handling Time

My team works to resolve online tasks assigned to them through a queue system; time taken to clear each task is measured (called handle_seconds). Approximately 18% of the tasks turn out to be ...
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### Compute conditional probability for Poisson binomial distribution

Consider $X=Y_{1}+\cdots+Y_{n}$, where $Y_{1}, \cdots, Y_{n}$ are $\mathrm{n}$ independent Bernoulli random variables with $Y_i\sim Bernoulli (1,p_i)$, $i=1,2,\cdots,n$. Then $X$ has a so-called ...
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### How to show linear combination of independent, but non-identically distributed Bernoulli's is asymptotically normal?

Summary I am curious about whether there exists theoretical justification to say a linear combination of a sufficiently large number of independent (but not identically distributed) Bernoulli random ...
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### Estimating variance of success probability in Poisson-binomial distribution

I am looking at a very large yet finite sequence of Bernoulli trials, each with its own probability. From the physical nature of the process, I know that the probabilities $p_i$ of each trial should ...
1 vote
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### Poisson process for the spatial analysis of accidents

I have a large dataset consisting of the geographic location, company, and date of accident. I also have a grid with a cell size that is 6 miles x 6 miles to disaggregate the data, since the dataset ...
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### Could laplace method be applied to discrete distribution such as bernoulli, poisson-binomial etc?

I am trying to reproduce the result of the essay: Variational inferences for partially linear additive models with variable selection(K. Zhao, H. Lian,2014)enter link description here and encounter a ...
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### How to efficiently estimate number of individuals with n+ successes from a series of bernoulli trials?

I have a situation where I need to estimate the number of persons exposed to a given event n or more times. For each person, I have an array of probabilities ...
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### Multiple coins with different but known bias: Probability of K heads with N coins and tosses [duplicate]

Suppose I have N biased coins. The bias of each coin $j$ is known: $p_j$. What is the probability that I throw at least K heads using all N coins and tossing them each once? The edge case of at least ...
1 vote
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### Conditional distributions of two mutually dependent binomial random variables

If $X,Y$ are mutually dependent binomial random variables, do we know how $Y|X$ and $X|Y$ are distributed? $X,Y$ are the sum of $n$ iid Bernoulli variables, \begin{align} X&=\sum_{i=1}^{n}X_i, \...
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### Simplify Equation with Random Variables

I'm wondering if whether the following problem has a solution. Suppose we have i random variables, all independent, and all following a Bernoulli distribution with parameter $p_i$ (all $p_i$'s are ...
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### Distribution of sum of possibly non-independent Bernoulli random variables with known variance-covariance matrix

I wonder if there are any results concerning the distribution of sums of possibly non-IID Bernoulli random variables when covariances in all pairs of r.v.'s are known. To make this more concrete ...
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### How to handle big count data with huge orders of magnitude in GLMMs: center & scaling but than negative values are introduced?

I'm relatively new to GLMMs and so far only handled relative data. Now I'm trying to model if the abundance of a taxon is affected in the disease state (condition) when considering random effects like ...
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### What is the name and formalism of this discrete distribution? [closed]

I am searching the name of something similar to a binomial distribution, but with individual probabilities (P(1) to P(N)). I calculated (brute-forced with a script) the probability of k positive ...
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### How to estimate probability of $\geq$ 1 success from a non-IID vector of probabilities, given many such vectors (now with asteroids)

I've got a deep neural net that returns sequences of probabilities. There are 25 probabilities per sequence. Many of these probabilities are zero, as a result of padding; when the input to the ...
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### MLE for the sum of independent Bernoulli trials with common factor

Suppose I am computing the sum of different bernoulli trials with probability $p_i = P s_i$, where $P$ is a common factor to all trials and $s_i$ is given, how can I compute the MLE for $P$? I realize ...
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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 ...
1 vote
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### Can I usefully apply the Lyapunov CLT condition to a finite sum of Bernoulli random variables? [duplicate]

I'd like to get a CLT-like approximate distribution (mostly tail behavior) of the sum $X$ of $n$ independent Bernoulli random variables $X_1, \dots, X_n$, with proportions $p_1, \dots, p_n$. The ...
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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 ...
1 vote
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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 ...