Questions tagged [compound-distributions]

When a random variable is distributed according to some parameterized distribution, where the parameter itself is a random variable. Also known as a "mixture" distribution, but the term "mixture" also has other senses in statistics.

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Joint Models - compound poisson

Let's say I have a longitudinal process that I can dicotomize as 0/1 depending on a literature established cutoff. I would be interested in modelling the number of events occurring for each individual ...
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Compounding Gamma with Gamma to yield F-distribution?

I am working through some problems from my Bayesian Statistics course and am having trouble understanding a step in the solution to a question. For reference this is the question: And here is the ...
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How does posterior predictive mean depend on parameters of the likelihood and prior distribution?

I have come across a problem in my research which deals with the mean of the posterior predictive distribution, i.e. $$p(x'|x)=\int d\theta p(x'|\theta)p(\theta|x)$$ where $x$ is an observed sample ...
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Which compound distributions of common discrete random variables have closed-form solutions?

I'm interested in modeling a number of processes that are chains of compounded distributions, i.e. $x \sim F(x|y)$, $y \sim G(z)$ etc., where the distributions $F$, $G$, etc. are common exponential ...
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How to specify a parameters for Gamma distribution?

I have a task: A frequency claim distribution, $K$ is a compound Poisson-Gamma distribution. The mean of the Poisson distribution is gamma distributed with mean equal to 1 and variation equal 2. Find ...
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Monte Carlo for Dirichlet Multinomial Model

Problem I am trying to implement Markov Chain Monte Carlo for the Dirichlet Multinomial mixture, described in this reference (where one used the expectation maximization algorithm). The model is as ...
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Find the characteristic function of a random variable involving a compoud poisson distribution

Let $\{\xi_i \}_{i=1}^{k}$ be a finite sequence of independent r.v. such that $\xi_{i} \sim F_i$. For each $i$, let: $\{\xi_{i,j}\}_{j=1}^{\infty}$ be a sequence of copies of $\xi_i$, that is: $\xi_{...
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What is the distribution of the Poisson Sum of gaussians?

I know that the sum of two independent normal random variables is normal. Particulary, when one is copy of other, i.e., if $X_1, X_2 \sim \mathcal{N}(0,\sigma^2)$, independent, we have: $$X_1 + X_2 \...
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Assumptions of compound Poisson model

My understanding of a compound Poisson RV is one defined as $$Y=\sum_{n=1}^N X_n$$ where $\{X_n\}_{n\in\mathbb{N}}$ is a sequence of identically distributed and mutually independent (iid) RVs $N$ is ...
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In the compound poisson distribution, when do I use that $X_j$ and $N$ are independent?

In the context of compound poisson distribution we have: $X \sim F$ a random variable; $\{X_n\}_{n\in \mathbb{N}}$ be a i.i.d. sequence of copies of $X$ - i.e. $X_j \sim F\,\, \forall \, j \in \...
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Is the sum (in some sense) of ARMA processes an ARMA process?

Given an ARMA(1,1) process $$X_t = \phi X_{t-1} + \varepsilon_t + \theta \varepsilon_{t-1},\quad \varepsilon_t \sim WN(0,\sigma^2)$$ Let $N \sim Po(\lambda)$ a poisson random variable. Consider the ...
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Using the method of moments or GMM to estimate the parameters of a specific problem

Given $(X_t)_{t \in \mathbb{Z}}$ an AR(1) process: $$X_t = c+ \phi X_{t-1} + \epsilon_t, \quad \epsilon_t\sim WN(0,\sigma^2)$$ We can show that $E(X_t) = \frac{c}{1- \phi}$ and $E(X_t^2) = \frac{\...
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Maximum likelihood estimate for multivariate sum of normal distributions

For each $j = 1,\dots,N$, let $\mu_j \in \mathbb{R}^N$ denote a known column vector, $\Sigma_j \in \mathbb{R}^{N\times N}$ a known covariance matrix, and $\theta_j \in \mathbb{R}$ an unknown parameter,...
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Decompound a Compound Probability Distribution

I am trying to figure out how to deconvolve or decompound a compound probability density function - knowing one of the distributions and having samples from the compound distribution. Assume I only ...
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Sampling Variance of Sample Proportion with a Discarded Outcome

Question: A random experiment with three exhaustive and mutually exclusive outcomes $A, B, C$ is performed $n$ times, resulting in a sample of outcomes $X_1, X_2, \ldots, X_n$. The following statistic ...
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Poisson Distribution Question- Viral Vector Integrations

The number of viral genomes that integrate in cells follows a poisson distribution (https://www.nature.com/articles/3302270). This assumes every target cell has the same infectivity. How does one ...
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Negative Binomial as Gamma-Poisson Mixture or Compound Logarithmic Poisson: can this correspondence be generalized to other distributions?

Preamble A random variable $X$ with a negative binomial distribution can be characterized in three ways: [Negative Binomial] $X\sim\operatorname{NegBin}(r,p)$ for some $r$ and $p$; [Gamma-Poisson ...
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Random forest on compound analysis and input data permutation

I am using a random forest model to associate climate variables with a specific type of impact, which is measured as the likelihood of failure (therefore, classification). The choice of random forest ...
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Simulation of compound Poisson Process with Lognormal jumps?

So I have the next problem: In order to simulate the ruin of a risk process I need, of course, to simulate the risk process itself but in this case this process has some characteristics that make it ...
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Compound beta-binomial and beta distribution

I have a process that is modelled by a beta-binomial, parametrised by mean $\mu$ and correlation $\rho = 1/(\alpha+\beta+1)$ (as per dbetabinom in the R VGAM package). I know $\rho$, but the mean $\...
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Estimate partially observed Poisson process

I try to estimate the intensity of a Poisson process $P_1$, but it is not fully observable. There are some "obervers" coming to the system which follow another Poisson process $P_2$. In ...
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Is this estimator biased or unbiased?

A random variable X constructed as follows: $$X = \sum_{i=1}^{N} Z_i \ $$ where $N$~Poisson$(\lambda)$ with $\lambda > 0,\space$ and$\space$ {${{Z_i}}$}$^N_{i=1}$ is an independent and identically ...
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If Z|Y ~ Bin(p, y) and Y ~ Poisson(L) then Z ~ Poisson(p*L)? [duplicate]

I checked whether this question was answered before but because of notation, it's hard to see. I am a reading a paper that defines the following two RVs $$ z \mid y \sim Binomial(\pi, y) \\ y \sim ...
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Poisson Distribution with Exponential Parameter

If we have $X(k)\sim Pois(2k)$ and $Y \sim Exp(15)$ and $Z=X(5Y)$. How can we determine $E(Z)$, $Var(Z)$ and $P(Z = z)$. So far I'm thinking $$\begin{align*} E(Z) &= E(X(5Y)) \\ &= E(Pois(10Y))...
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Looking for the name of a distribution and/or a way to calculate its cumulative distribution function

For parameters $\nu_1$, $\nu_2$, $\lambda$, define random variables using a chi-squared distribution and a non-central chi-squared distribution: $$ S \sim \chi^2_{\nu_2} / \nu_2 $$ $$ F \sim \frac{\...
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10 votes
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Poisson Gamma Mixture = Negative Binomially Distributed?

This paper introduces a model called "Beta-Geometric / NBD" which models "repeat-buying behavior in settings where customer “dropout” is unobserved: It assumes that customers buy at a ...
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finding PDF of Y, given Y|X [closed]

$$Y|X\sim Bin(X,n)$$ $$X\sim U([0,1])$$ How can I find the PDF of Y? I know that: $$\Bbb P(Y=k)=E_X[\Bbb P(Y=k)|X]$$
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Stratified sampling / QMC simulation for compound Poisson rv

I have a rv $X$ of the form $$ X=\sum_{i=1}^N Y_i, $$ where $N$ is a discrete rv (often, but not always, Poisson) and $Y_1,\ldots,Y_N$ are continuous random variables, iid and independent from $N$. I ...
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2 answers
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Parameter estimation for random variables where a control parameter is another r.v

Let $\{X_i\}$ a sequence of independent random variables. Each $X_i$ has a p.d.f $p(m, \theta)$. Where $\theta$ is a real unknown parameter and $m$ the outcome of another random variable $M$ with p.d....
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What is the distribution of a mixture of exponential distributions whose rate parameters follow a gamma distribution?

I want to know the theoretical distribution of a mixture of exponential distributions whose rate parameters are distributed according to a gamma distribution: $$ y\sim\text{Exp}(\theta), \quad\text{...
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2 votes
1 answer
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Mean and variance and special distribution of events

A hospital Accident and Emergency (A&E) department receives an average of 6 ambulances an hour. It can process patients in 30 minutes, but if it receives more than five patients in 30 minutes, ...
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2 answers
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Given distribution of $X$ and $X|Y=y$, is it possible to find distribution of $Y$?

What the title says! My intuition is NO since in Bayesian statistics we typically specify the prior and likelihood, and from those two we can compute the posterior and so on. We can interpret $Y$ = ...
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Compound distribution

I'm trying to compute a maximum likelihood of compound Poisson exponential distribution in R by using EM-algorithm method. The distribution is defined by $∑N_j=1 Y_j$ where $Y_n$ is i.i.d sequence ...
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Compounding a gamma distribution with another distribution to yield a gamma

I have a gamma distributed random variable $X$, with its mean $\mu$ distributed as some other function $$ X \sim \text{Gamma}(\mu,k)\\ \mu \sim P(\theta) $$ What is the distribution $P(\theta)$ such ...
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How to find an expression of the variance of a Poisson-Lognormal distribution?

I am using a model for the number of goods in a supermarket cart with a Poisson-lognormal distribution (a lognormal mixture of Poissons). I would like to find an expression of the variance of this ...
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Derivation for mixed distribution, Poisson-Lindley

I want to derive the Poisson Lindley Distribution. $$ f_x(x|\lambda) = \frac{\lambda^{x-1}}{(x-1)!}e^{-\lambda} $$ $$f_x(x|p) = \frac{p^2}{(p+1)}(\lambda+1)e^{-\lambda p} $$ The Distribution of x, $...
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Compounding a Gaussian distribution with variance distributed according to the absolute value of another Gaussian distribution

Have there been earlier descriptions of the following compound distribution? Compounding a Gaussian distribution with variance distributed according to the absolute value or square of another ...
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3 votes
1 answer
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Mean of a Poisson-Lognormal Distribution (PLN)

I would like to calculate the mean value of a PLN distribution, $$ f(x;\mu,\sigma)=\frac{1}{x!\sigma\sqrt{2\pi}}\int_{0}^{\infty}\lambda_\ast^{x-1} e^{-\lambda_\ast} e^{-\frac{(log(\lambda_\ast-\mu)^2}...
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When can a finite mixture of distributions drawn from a distributional family be well-described by a distribution from the same family?

This question is motivated by a situation that we frequently encounter in economic variables. We believe that the overall distribution of some variable is well approximated by a particular ...
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1 answer
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Binomial distribution for randomly drawn probabilities

Setting Probability theory can be a weird place sometimes. Here I was, confident in my insane math skills, trying to solve the following problem: Let $N, \alpha$ and $\beta$ be given. ...
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Compound poisson distribution [closed]

I'm trying to compute a maximum likelihood of compound Poisson gamma distribution in R. The distribution is defined by $ \sum_{j=1}^{N} Y_j $ where $Y_n$ is i.i.d sequence independent $\operatorname{...
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1 vote
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Compound Poisson process for demand

I have a demand pattern for a service part. Demand event rate of this part is Poisson distributed. Demand of the part is 3 times in a year. So the demand event rate is 0.25/Month. Each demand ...
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What is the distribution of a Poisson variable, where the Poisson rate is Normal (or Binomial)?

What is the distribution of $X$ if $$ X \sim \text{Poisson}(\lambda), \quad \text{where }\lambda \sim N(\mu,\sigma^2)$$ or $$ X \sim \text{Poisson} (\lambda), \quad \text{where }\lambda \sim ...
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6 votes
1 answer
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Compound Poisson random variable

A compound Poisson random variable $S$ is defined as: $S=\displaystyle\sum^N_{i=1}X_i,$ where $N$ is a random draw from a Poisson distribution with intensity parameter $\lambda$, and $X_i$ are ...
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8 votes
2 answers
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Characteristic Function of a Compound Poisson Process

The definition of a compound Poisson process and its characteristic function I have are the following: Let $\lambda>0$ and $N\sim\text{Poisson}(\lambda T)$. Also, $\{X_i\}_{i=1}^N$ are i.i.d. ...
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2 votes
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Compound Distributions --- Basic Techniques and Key General Results from First Principles

Could someone please point me to a source with notation, terminology, key results and basic techniques to approach compound distributions? Definition Compound probability distribution is the ...
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Compound Distribution --- Uniform Distribution with Normally Distributed Parameters

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Uniform Distribution whose parameters are distributed ...
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3 votes
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PMF of compound Poisson process?

Can I obtain an analytic expression for PMF of compound Poisson process? $Y_t = \sum \limits_{i=1}^{X_t} D_i$, where $X_t \sim \mathcal{Poisson}(\lambda)$ and $D \sim \mathcal{Geometric}(\rho)$.
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2 votes
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Can I estimate Variance of Gamma from Negative Binomial distribution distributed data, given NB is Gamma-Poisson compound

I believe the data I have follows Negative Binomial distribution (over-dispersed Poisson). We know Negative Binomial is a Gamma-Poisson compound distribution. The variance of this Gamma distribution ...
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Compendium or catalog of compound distributions?

Does anyone know of a good compendium or catalog of compound distributions, or finite mixture representations of those distributions? I am trying to find out to what extent the common multi-...
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