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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Compound binomial distribution distributed as binomial

Suppose we have independent family of random variables $\{Y\}_{i\in\mathbb{N}}\cup\{N\}$, where $Y$s are identically distributed. Next consider a sum of random number of random variables $W_N\equiv\...
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A question about the skewness of a compound Poisson distribution

I'm trying to intuitively answer the last part of item b of question 21 of chapter 9 (Mixtures and Compound Distributions) of the book Probability: the science of uncertainty with applications to ...
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Which is $E[L]$ and $Var(L)$ in Compound Poisson (in terms of F function)?

I have a compound Poisson model that calculates the danger to my business. I know that the loss to my business in a time interval $T$ is $L = \displaystyle\sum_{i=1} ^{N(T)} X_i$ with $N(T) \sim Poi(\...
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Is every Compound Poisson distribution a Mixture model?

We have two models: Let $N \sim \hbox Poisson (\lambda)$ and let $(X_k ; k =1,2,3,...)$ be a a sequence of independent and identically distributed random objects (random variables, vectors or ...
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An estimation problem related to a certain stochastic process (radom sum of stochastic processes)

Let $N\sim \hbox{Poisson}(\lambda)$ and $(X_t)_{t \in \mathbb Z}$ be a stochastic process. Consider the following stochastic process: $$ Y_t = \sum_{j=1}^N X_{t,j}$$ where $(X_{t,1})_{t\in \mathbb Z}, ...
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A question about conditional expectation

Let $X_1,X_2,...,X_n$ and $Y$ be random variables. I know that: $$\label{aaa}\tag{I} E\left[\sum_{j=1}^n X_j \Bigg | Y \right]=\sum_{j=1}^n E\left[ X_j \Big | Y \right] $$ Now, suppose that $Y$ ...
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conditional probability of compound Poisson process

Suppose that $X_1, X_2, \dots $ are identically independent random variables following a distribution function $F$. Let $N(t)$ follow a Poisson process with intensity parameter $\lambda$. Then for $...
2 votes
1 answer
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Trying to understand if a model is an empirical Bayes or not

I am trying to understand if the model published in section 4.1.1 here is or is not an Empirical Bayes model (which the author claims it is). Or, maybe, if it is a valid one or not. The model looks as ...
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Simulating a particular non-ergodic process

I'm trying to visualize by some simulations the non-ergodicity of a stationary process, but I'm not getting it. I'm interested in the following particular case. Let $(X_t)_{t \in \mathbb N}$ be a ...
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What is the distribution of the sum of a normal and two compound random variables?

Let $Y_1 \sim CP(\lambda_1, d F_Z ) $ and $Y_2\sim CP(\lambda_2, d F_W )$ be two Compound Poisson random variables and $X = \mathcal{N} + Y_1 + Y_2$ with $\mathcal{N}\sim N(0,1)$ such that $\...
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How to calculate Average CAGRs?

I have sales by 30 different products over years (2015-2022). I need to compute Compound Annual Growth Rate, however some of the product groups have high CAGR and some of them have quite low CAGR. So ...
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How can compound a exponential distribution with gamma distribution?

A random variable X is said to have a exponential distribution with parameter lamda > 0. if its pdf is given by f(x,lamda)=(lamda * exp(-lamda * x)). A random variable X is said to have a gamma ...
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Binomial distribution conditional on previous sampling

When using a Binomial distribution, the bulk of literature assumes that the probability of success $p$ is known, or deterministic. I would like to get the distribution for a Binomial RV with unknown ...
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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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1 vote
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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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1 vote
1 answer
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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 ...
7 votes
2 answers
636 views

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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2 votes
1 answer
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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 ...
1 vote
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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 $\...
1 vote
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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 ...
1 vote
3 answers
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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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4 votes
3 answers
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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{\...
10 votes
2 answers
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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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1 vote
1 answer
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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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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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4 votes
2 answers
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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{...
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, ...
2 votes
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 ...
2 votes
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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 ...