# 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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### Expected value after $K$ Bernoulli trials where the $i$-th probability of success depends on the current number of successes

I have an experiment that involves $K$ Bernoulli trials. Trial $i$ has probability of success $p_{i, n}$ where $n$ is the current number of successes (so $0 \leq n \leq i-1$). If my random variable is ...
22 views

### How to report negative binomial results with a multi-level categorical variable?

I would like to ask 2 questions: the first, as indicated in the title, concerns how to report the results of the 'negative binomial model'. The second, differently, relates to how to interpret the ...
1 vote
52 views

### What is the distribution of a Poisson-Binomial variable where the probabilities of success are from another distribution?

If I have a Poisson-Binomial random variable $X$ built from $n$ trials where I draw each $p_i$ as either $a \in \left[0, 1 \right]$ or $b \in \left[0, 1 \right]$ with equal probability. How can I find ...
90 views

### Confidence interval for sum of independent but not identical bernoulli RVs with small sample size

I have a small sample size (5 <= N <= 10), and for each sample i, we observe independent $Y_{i}$ where $Y_{i}$ is the sum of 7 independent yes/no responses (i.e. bernoulli experiments), where ...
47 views

### Is it possible to efficiently compute the probability of k+ events in a Poisson Binomial Distribution?

I have a process that is currently being effectively modelled by a Poisson Binomial distribution (wiki link). We have access to all of the constituent probability values of the independent trials; ...
19 views

### R implementation of a Multinominal Problem: Probability of n-times head in k throws with varying probabilities per throw

im struggling with a potential easy to solve Problem. I have a dataset with 100k series of coin throws with varying k (throws). For each series I want to compute the the probability for each discrete ...
47 views

### What is the Poisson binomial probability, for one flip of one fair coin and two unfair coins with probability-of-heads $p_u$, of flipping $r$ heads?

The binomial probability, for one flip of $n$ unfair coins with probability-of-heads $p_u$, of flipping $r$ heads $$B(n, r, p) = C(n, r) \ p^r \ (q = 1 - p)^{n - r}$$
92 views

### Is the OLRE term meaningful in the negative binomial model? + Is overdispersion in the NB model an issue?

I'd like to ask three questions regarding the negative binomial (NB) regression / distribution. The NB model with NB2 parameterization ($var(Y_{NB2}) = \mu + \frac{\mu^2}{\theta}$) is sometimes ...
1 vote
67 views

### 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 $A$, $B$ ...
27 views

### 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 ...
187 views

### 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 ...
226 views

### 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 ...
1 vote
93 views

### 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 ...
70 views

### 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 ...
115 views

### 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 ...
1 vote
86 views

### 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
505 views

### 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, \...
1 vote
46 views

### 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 ...
495 views

### 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 ...
1 vote
122 views

### 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 ...
1 vote
42 views

### 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 ...
1 vote
76 views

### 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 ...
103 views

### 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 ...
54 views

### 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 ...
434 views

### 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 ...
2k views

### 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 ...
248 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 ...
1 vote