Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [poisson-binomial-distribution]

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

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

The binomial distribution is a special case of the Poisson binomial distribution.

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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 $\Pr(X_i=1)=p_i$ and $\Pr(X_i=0)=1-p_i$. I am…
David B
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Sum of Bernoulli variables with different success probabilities

Let $x_i$ be independent Bernoulli random variables with success probabilities $p_i$. That is, $x_i=1$ with probability $p_i$ and $x_i=0$ with probability $1-p_i$. Is there a closed expression or an approximate formula for the distribution of the…
becko
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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 a better way of calculating these probabilities…
Maha123
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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 positive and between ~0.001-1,000 with a handful of…
Leon P
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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 $X_i\sim Ber(p_i)$. The Poisson-Binomial…
cangrejo
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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 example, 10 workers. Each worker can produce $N$…
Boris Mikhaylov
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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 data for training?
Łukasz Lew
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Hypothesis testing on Poisson Binomial distribution

Imagine $N$ baskets each filled with a different number of products chosen from a set $P$. It is possible to have repeated products in the same basket. For instance: basket 1 = [A,B,B,C] basket 2 = [A,B,C,D,F,J] basket 3 = [C] ... basket N =…
alberto
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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 many (up to one trillion, $10^{12}$) unique,…
Thomas Mueller
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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 says when the $p_i$ are small the distribution…
Chris Jones
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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 other words, the Bernoulli trials have different…
anonymous
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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 graph", this is the first image result I…
derekchen14
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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, the distribution of $X$ is called the Poisson…
Ian Hincks
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Estimate accuracy of an estimation on Poisson binomial distribution

I manage a website that charges its customers using payment cards. Some transactions area approved, others are declined. I compute the approval rate of transactions for a interval (a calendar day) as the number of approved transactions (A) divided…
axiac
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Probability mass function with variable probability?

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. I found that I can use PMF when I have the same…
Jorge Riv
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