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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Discrepancy between coefficient and mean difference in predicted values of logistic regression
I'm using a poisson-binominal logsitic regression model to analyze a list experiment (item count technique) where the outcome variable is a binary response of the respondent to a sensitive item (e.g., ...
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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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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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Logistic / multinomial regression as two / multiple Poisson regressions?
Can we instead of doing logistic or multinomial regression do two or multiple Poisson regressions and then combine Poisson predictions to get probabilistic predictions? If yes, how should we transform ...
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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 ...
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Confused between Multiple Random Variables and Likelihood Function [closed]
I am confused between the two at a very fundamental level. Following is the problem:
I take observations $\vec{x}$ and create a histogram $\mathbf{n} = (n_1,\ldots,n_N)$ out of it with $N$ bins. ...
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Conditions for Binomial Distribution
It is known that if $X$ is the sum of $n$ independent and identical Bernoulli random variables, $X$ follows a Binomial distribution.
How about the reverse, can a sum of dependent and/or non-identical ...
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Confidence interval for the sum of 2 binomially distributed variables
$P_1$ and $P_2$ are uncorrelated, binomially distributed variables with success probabilities $p_1 \neq p_2$. Say I measure:
$k_1 = 9$ successes out of $n_1 = 10$ trials for $P_1$ and
$k_2 = 1000$ ...
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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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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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Distribution of a sum of a product of Bernoulli vectors divided by the sum of the first Bernoulli vector
I'm trying to understand the distribution of a normalized sum of a product of Bernoulli random variables. Specifically, I have two vectors $M_1$ and $M_2$ of length n. Each element of each vector is a ...
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Probability that sum of binary variables is even
Let $S_i \in \{0,1\}$, $i=1,\dots,N$ be $N$ independent random binary variables, each taking the value 1 with probability $0 \le p_i \le 1$ (and the value 0 with probability $1-p_i$).
I am interested ...
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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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Estimating variance of Poisson Binomial random variable
Let's say I have a weighted coin, with probability $p_i$ of being heads. I flip $N_i$ times, and estimate $P_i$ and the variance on $p_i$ using the relevant formulas for a Binomial distribution. Call ...
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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 ...
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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 ...
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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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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 ...
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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 ...
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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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How can I efficiently model the sum of Bernoulli random variables?
I am modeling a random variable ($Y$) which is the sum of some ~15-40k independent Bernoulli random variables ($X_i$), each with a different success probability ($p_i$). Formally, $Y=\sum X_i$ where $\...
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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 ...
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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; ...
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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 ...
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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
$$B(n, r, p) = C(n, r) \ p^r \ (q = 1 - p)^{n - r}$$
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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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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 $k<\sum_{i=1}^{n}...
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Modelling probabilities of a sum of binomials with different probabilities and trials
I have the following example data, where each row is an independent observation:
A
B
C
Y
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$A$, $B$, $C$ and $Y$ are all positive integers. The variables $A$, $B$ ...
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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 ...
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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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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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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 ...
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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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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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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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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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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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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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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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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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Modified Poisson binomial distribution
In the Poisson binomial distribution each trial is either yes (1) or no (0). Is there a distribution where each trial is either yes (N) or no (0)?
I'd like to model a situation where I have, for ...
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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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