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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1 answer
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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 ...
2 votes
1 answer
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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 ...
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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 ...
3 votes
1 answer
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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 ...
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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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1 answer
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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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1 answer
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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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2 votes
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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 ...
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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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152 views

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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Lower bounds for Poisson Binomial distribution tail probability

Consider a Poisson Binomial distribution $X = X_1+X_2+\dotsm+X_n$, 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}...
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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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1 answer
290 views

Poisson binomial distribution hypothesis test

Let $X_i$, $i=1, \dots, n$, be independent non-identically distributed random variables with Bernoulli distributions with unknown probability of successes $p_i$, $i=1, \dotsc, n$. Then $Y:=\sum_{i=1}...
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Is the mode of a Poisson Binomial distribution next to the mean?

A Poisson-Binomial variable $X\sim PB(p_1, \dots, p_n)$ is the sum of $n$ independent, not necessarily identically distributed, Bernoulli variables $X_1, \dots, X_n$: $$ X=\sum_{i=1}^n X_i, $$ with $...
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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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3 votes
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Poisson Binomial Distribution - confidence intervals for sum of unequal probabilities estimated with uncertainty

I believe my question is related to, but distinct from this one: Poisson Binomial Distribution - confidence intervals I am working to estimate species richness by summing the results of 12 individual ...
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1 answer
423 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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1 answer
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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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Correlation coefficient of x and y

If we have $$ X\sim Poisson(\lambda), Y|X = x\sim Binomial(x+1,p) $$ What is the correlation coefficient of X and Y? So I used $$\rho=\frac{Cov(X,Y)}{\sqrt{Var(x)Var(Y)}} = \frac{E[X[E[Y|X]]-E[X]E[...
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Probability of at least one success in a series of independent, non-identical Bernoulli trials

Let's say I have a set of independent Bernoulli trials each with a different probability: $$ x_i \sim \operatorname{Bernoulli}(p_i) $$ The number of successes (sum of x) will be distributed ...
2 votes
1 answer
928 views

Poisson Binomial Distribution - confidence intervals

I'm working on a project which involves multiple trials for which the probability of success is not the same across trials. Given the unequal probabilities per trial, I'm using the Poisson Binomial ...
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2 votes
1 answer
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How to model a recursive probabilistic experiment?

I have a theoretical experiment as follows: We have two boxes A and B, and N unfair coins (each has different probability for showing Heads). At the beginning, box A contains all N coins and box B ...
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1 vote
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Which kind of diagnostic plots for count data? [duplicate]

I know that for an lm model is enough to run plot(model_lm) to get diagnostic plots. I am dealing with high-dimensional count ...
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2 votes
2 answers
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Expected value of $e^{sP}$ where s is a complex number and $P$ is a Poisson rv

For each positive integer $N$, let $ B_N$ be a binomial $(N,1/3)$ random variable and $P$ be a Poisson(5) random variable. I am trying to understand the statistics of $B_P$. Could someone please hint ...
4 votes
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Limit behavior of "weighted" Poisson Binomial distribution

Given $X_n \sim \operatorname{Binomial}(n, p_n)$ it is known by the Poisson Limit Theorem that as $n \to \infty$ that $$X_n \to \operatorname{Poisson}(np_n).$$ This can be generalized to hold for ...
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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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2 votes
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Summation of Series involving Exponential terms

I'm currently working on a problem, which involves Poisson-Binomial Distribution. https://en.wikipedia.org/wiki/Poisson_binomial_distribution . The Mean of PBD is given by $M=\sum_{i=1}^{n}p_i$ ....
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1 answer
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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 (...
2 votes
1 answer
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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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4 votes
1 answer
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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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3 votes
0 answers
679 views

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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1 vote
1 answer
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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 ...
2 votes
0 answers
86 views

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 ...