# 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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### 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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### How to fit a compound poisson distribution based on a small dataset?

Suppose I have this small data set and it is required to fit a compound poisson distribution based on the data [with respect to the the real blood corpuscular (x) per cell] available. \begin{array}{| ...
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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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### A Binomial distribution of which the number of trials is a Poisson distribuion [duplicate]

It is known that X is a random variable which follows Poisson distribution Po(λ), Y is a random variable which follows Binomial distribution Binomial(X,p). The problem I encountered is: Proof that Y~...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### Binomial distribution where probability of success is dependent on another binomial distribution

How does one model the Binomial distribution where the probability of success is the result of another Binomial distribution. For example, say I make 10 coin tosses many times and record the number ...