Questions tagged [poisson-binomial]

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

Filter by
Sorted by
Tagged with
0
votes
0answers
6 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 ...
2
votes
0answers
34 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 ...
2
votes
1answer
53 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 ...
2
votes
1answer
29 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 ...
2
votes
0answers
85 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 ...
0
votes
0answers
22 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 ...
1
vote
0answers
32 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 $k<\sum_{i=1}^{n}p_i$. ...
0
votes
0answers
48 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 ...
3
votes
1answer
73 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}...
7
votes
2answers
299 views

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 $...
2
votes
1answer
96 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?
1
vote
0answers
13 views

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 ...
0
votes
1answer
72 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 ...
0
votes
1answer
106 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, ...
0
votes
1answer
274 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)...
0
votes
1answer
64 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[...
0
votes
1answer
233 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 ...
2
votes
1answer
419 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 ...
2
votes
1answer
68 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 ...
1
vote
0answers
47 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 ...
2
votes
2answers
76 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 ...
3
votes
0answers
73 views

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 ...
0
votes
1answer
436 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 ...
2
votes
0answers
189 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 ...
0
votes
2answers
44 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 ...
0
votes
0answers
161 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 ...
1
vote
1answer
612 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 (...
2
votes
1answer
359 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 ...
4
votes
1answer
94 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. ...
3
votes
0answers
394 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 ...
1
vote
1answer
885 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 ...
2
votes
0answers
75 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 ...
2
votes
0answers
98 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 ...
2
votes
0answers
226 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 = ...
2
votes
1answer
260 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 ...
5
votes
2answers
456 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 ...
4
votes
1answer
156 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 ...
9
votes
4answers
3k views

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 ...
0
votes
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 ...
3
votes
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 ...
2
votes
0answers
196 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 ...
2
votes
0answers
515 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 ...
1
vote
1answer
141 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 ...
1
vote
1answer
126 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 ...
2
votes
1answer
380 views

Bounding tail probabilities of the Poisson Binomial Distribution?

In a problem I'm working on, I have Bernoulli random variables $X_1,X_2,\dots,X_k$ ($k$ is odd) and I am interested in their sum $Y = \sum_{i=1}^k X_i$. In this problem, $P(X_i=1) = p_i$ and $P(X_i=0) ...
3
votes
3answers
1k views

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 ...
1
vote
2answers
427 views

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? ...
3
votes
2answers
2k views

Success of Bernoulli trials with different probabilities and without replacement

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 ...
2
votes
1answer
233 views

Efficiently computing poisson binomial sum

Suppose there is a match between two teams where the first team to win a certain number of games wins the match. The match is handicapped, with Teams A needing to win $H_A$ games and Team B needing $...
0
votes
2answers
462 views

A Tail Bound For Poisson Binomial Distribution?

Consider the Poisson-Binomial Distribution with two components. Let $Y_0\sim bin(n,p_0)$, $Y_1\sim bin(n,p_1)$, and let $Y=Y_0+Y_1$. For any $k>n(p_0+p_1)$, Can we upper bound the tail probability ...