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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27 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 ...
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8 views

Regression of bounded count variable, with zeroes, and non-independent trials

I'm looking for a way to model 'days alive and out of hospital' after surgery (see link). The variable is basically a count of patient status on each followup day, but is quite badly behaved, ...
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45 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 ...
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36 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 ...
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9 views

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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38 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 ...
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57 views

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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1answer
82 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 ...
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1answer
82 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 ...
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101 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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37 views

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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51 views

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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59 views

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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1answer
112 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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183 views

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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1answer
169 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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1answer
212 views

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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1answer
419 views

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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67 views

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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385 views

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 ...
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1answer
542 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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1answer
71 views

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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63 views

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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2answers
77 views

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 ...
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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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1answer
537 views

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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195 views

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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2answers
45 views

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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184 views

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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30 views

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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1answer
705 views

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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1answer
513 views

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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1answer
100 views

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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450 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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1answer
1k views

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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77 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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113 views

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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242 views

How to derive posterior distribution for non-informative Gamma prior distribution?

Lets assume I have 1000 cases of cancer occurring per year in Barcelona. How would I proceed to estimate/derive posterior distribution for this data? I know that posterior defined as: $Posterior = ...
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1answer
313 views

Are binomial regression and Poisson regression with an offset to 1 substantially different?

I know that negbin can approximate the betabin distribution, especially when the probability of hitting the max is low (events are more rare). If the offset of a negative binomial regression ...
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2answers
511 views

Random number generator that returns unique 64-bit numbers in sorted order

(I have asked the question at stackoverflow.com here but maybe it's better to ask here for the statistical part. Feel free to correct my question, fix the tags, redirect me...) I need a generator for ...
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1answer
158 views

Poisson Binomial Distribution with Evenly Distributed Bernoulli Trial Probabilities

Consider the Poisson binomial distribution with $n$ coins and coin probabilities ${1 \over n}, {2\over n}, \dots, {n-1 \over n}, 1$. Do we know an asymptotic for this distribution? Le Cam's theorem ...
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4answers
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What is the CDF of the sum of weighted Bernoulli random variables?

Let's say we have a random variable $Y$ defined as the sum of $N$ Bernoulli variables $X_i$, each with a different, success probability $p_i$ and a different (fixed) weight $w_i$. The weights are ...
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1answer
88 views

Is using “Normal Approximation to binomial distribution” to test mutation enrichment in genomic region correct?

We are analyzing cancer patient mutation data. We defined set of region on the human genome as binding events, (for the ones who is interested in to the subject, it is a transcription factor binding ...
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2answers
50 views

How to test if turtles are dying mostly at a particular time of year?

I have citizen science records of turtle sightings (alive/dead, and if dead what caused its death e.g. car/predator/etc), and the date they were spotted. What would be the most appropriate test to ...
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0answers
215 views

Hypothesis testing on Weighted Poisson Binomial Distribution

Suppose I have $i$ coins, all of which are weighted to have a different probability $p$ of flipping heads. This results in $i$ Bernoulli distributions with different $p_i$. Cumulatively, this results ...
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572 views

Multinomial distribution with different probabilities for each trial

If in a binomial distribution, the Bernoulli trials are independent and have different success probabilities, then it is called Poisson Binomial Distribution. Such a question has been previously ...
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1answer
187 views

Conditional pdf on a Poisson's Binomial Distribution

Suppose I have a certain nEV number of Electric Vehicles, each one has a State of Charge (SoC) picked from a particular distribution X. I have a certain quintic polynomial function g(x) which maps ...
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1answer
141 views

Poisson binomial distribution-like problem

Given n trials, where, on each trial, you have a given probability of either winning or losing a set amount of money (with both the amount of money and the probability changing for each trial)- what ...