Questions tagged [poisson-binomial]

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

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6
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2answers
259 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 $...
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1answer
24 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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12 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 ...
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1answer
43 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
52 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
92 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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1answer
46 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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1answer
99 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
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1answer
226 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
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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 ...
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0answers
28 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
70 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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48 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 ...
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1answer
257 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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141 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
43 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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89 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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1answer
387 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 (...
4
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1answer
83 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. ...
2
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329 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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0answers
72 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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76 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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208 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
190 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
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2answers
403 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
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1answer
132 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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3answers
2k 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 ...
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1answer
87 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
47 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
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0answers
157 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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446 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
106 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
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1answer
336 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
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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 ...
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2answers
378 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
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2answers
1k 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
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1answer
176 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 $...
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2answers
383 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 ...
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0answers
27 views

determining metric with largest excursion

I'm trying to use an algorithm to determine which of many metrics is the one with the largest excursion value from a baseline dataset. For simplicity, let's say I have two measurements. MetricA has a ...
5
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1answer
697 views

Maximum likelihood estimation of a Poisson binomial distribution

According to Wikipedia, the Poisson binomial distribution is the discrete probability distribution of a sum of independent Bernoulli trials that are not necessarily identically distributed In ...
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0answers
58 views

Efficiently draw variates from Poisson-Binomial distribution

What is the best way to sample from a Poisson-Binomial distributed random variable $Z$? Since $Z = \operatorname{Bern}(p_1) + \operatorname{Bern}(p_2) + ...$ we could sample from each Bernoulli ...
4
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1answer
819 views

Skewness for a sum of independent weighted bernoulli random variables with different probabilities of success

Suppose $Z_i$ are independent Bernoulli random variables with differing probabilities $P_i$. Also suppose weights $W_i$ are positive and constant. Let's define the random variable $S$ which is the ...
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0answers
126 views

Comparing the rates of Poisson distribution using Bayesian inference

In the 'Theory of Probability' book by Sir Harold Jeffreys, (5.15), the form of the Bayes Factors that he derives for the comparison of Poisson rates is the same as that of Binomial rates. But I did ...
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0answers
33 views

Time Series/Predicting discrete values from a finite population

I'm at a complete loss as to how to approach this real-world problem. We have customers that are submitting requests to upgrade their service. I'm trying to create a model that can predict both the ...
11
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2answers
6k views

Success of Bernoulli trials with different probabilities

If 20 independent Bernoulli trials are carried out each with a different probability of success and therefore failure. What is the probability that exactly n of the 20 trials was successful? Is there ...
5
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1answer
6k views

Why is a binomial distribution bell-shaped?

I would expect there to be only be values between zero and one (with 0 => failure and 1 => success), but instead the values go up much higher. For example, if I search for "binomial distribution ...
4
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1answer
749 views

Variance of the Poisson Binomial Distribution

Consider a sequence of $n$ independent Bernoulli trials drawn from a list of biases $p_1,p_2,...,p_n\in[0,1]$, respectively. We set the random variable $X$ to be the sum of these trials. On wikipedia, ...
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2answers
1k views

Expected number of successes from $N$ Bernoulli trials with different $p$

Suppose I have N probabilities $(p_1, p_2,...,p_N)$ that represent the chance that each that a corresponding test was passed. How do I apply the Bernoulli distribution to determine the expected number ...
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1answer
110 views

Inferential statistics for vector of percentages

I'm getting confused by this and was wondering if someone can enlighten me: I have a random sample consisting of 50 percentages. Each percentage can take on any value between 0% and 100% inclusive ...
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0answers
3k views

Poisson Distribution Analysis in SPSS - Nonparametric, count, repeated-measures data

In relation to a recent post about what analyses to conduct for a data set, I am now asking a related question about the test to run in SPSS. Background information on the data: Repeated measures ...