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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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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479 views

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 ...
• 103
121 views

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 $k_2 = 1000$ ...
• 41
1 vote
63 views

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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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 ...
• 153
131 views

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 ...
• 227
64 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 ...
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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 ...
• 227
145 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 ...
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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; ...
• 308
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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 ...
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 $$B(n, r, p) = C(n, r) \ p^r \ (q = 1 - p)^{n - r}$$
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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 ...
1 vote
71 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 $A$, $B$ ...
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32 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 ...
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212 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 ...
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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 ...
1 vote
109 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 ...
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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 ...
• 131
119 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 ...
• 308
1 vote
113 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 ...
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1 vote
566 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, \...
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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 ...
612 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 ...
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1 vote
135 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 ...
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1 vote
44 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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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 ...
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115 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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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
87 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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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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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 ...
276 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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1 vote