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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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10 votes
4 answers
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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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4 votes
2 answers
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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 $k_2 = 1000$ ...
dimitsev's user avatar
1 vote
1 answer
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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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25 votes
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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 ...
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5 votes
2 answers
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 ...
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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 ...
Ric87's user avatar
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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 ...
Germ's user avatar
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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 ...
David D's user avatar
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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 $$B(n, r, p) = C(n, r) \ p^r \ (q = 1 - p)^{n - r}$$
Tom Lever's user avatar
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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 ...
Eva Šragová's user avatar
1 vote
1 answer
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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 10 22 6 2 4 60 2 0 12 8 10 3 ... $A$, $B$, $C$ and $Y$ are all positive integers. The variables $A$, $B$ ...
duncster94's user avatar
0 votes
1 answer
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 ...
geoabram's user avatar
3 votes
1 answer
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 ...
John Stone's user avatar
3 votes
1 answer
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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 ...
user365119's user avatar
1 vote
1 answer
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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 ...
driyg's user avatar
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2 votes
1 answer
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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 ...
Josh's user avatar
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1 vote
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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 ...
broidul's user avatar
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1 vote
0 answers
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, \...
arb's user avatar
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0 answers
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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 ...
damianodamiano's user avatar
4 votes
1 answer
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
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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
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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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1 vote
1 answer
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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 ...
generic_user's user avatar
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2 votes
0 answers
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 ...
WillRB's user avatar
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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 ...
Anonymous Scientist's user avatar
1 vote
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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 ...
xmq's user avatar
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4 votes
1 answer
480 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 ...
bob's user avatar
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5 votes
2 answers
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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 ...
cookie monster's user avatar
2 votes
0 answers
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 ...
MrCorote's user avatar
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1 vote
0 answers
241 views

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}...
Shubham Jain's user avatar
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0 answers
153 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 ...
Alex Darsonik's user avatar
3 votes
1 answer
504 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}...
Matěj's user avatar
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9 votes
2 answers
646 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 $...
cangrejo's user avatar
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3 votes
1 answer
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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?
Pie-ton's user avatar
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1 vote
0 answers
70 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 ...
jvriesem's user avatar
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3 votes
0 answers
351 views

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 ...
user2870897's user avatar
0 votes
1 answer
688 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 ...
Elliott de Launay's user avatar
3 votes
1 answer
2k 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, ...
ZUN LI's user avatar
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0 votes
1 answer
1k 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)...
user1673216's user avatar
1 vote
1 answer
135 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[...
Immanuel Kunt's user avatar
0 votes
1 answer
1k 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 ...
snakeoilsales's user avatar
3 votes
1 answer
1k 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 ...
KirkD_CO's user avatar
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2 votes
1 answer
104 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 ...
Angie's user avatar
  • 153
1 vote
0 answers
88 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 ...
Mary's user avatar
  • 11
2 votes
2 answers
129 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 ...
pickle_lover's user avatar