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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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Need help in calculating $\mathbb{E}(\frac{1}{x_{(2)}-x_{(1)}}\int_{x_{(1)}}^{x_{(2)}} f(t) \ dt)$, where $x_{(i)}$ are related Beta distribution

Suppose $Y, Z \stackrel{\text{iid}}{\sim}\mathrm{Uniform}(0,1)$. Let $a = g(\min(y,z)),\ b=g(\max(y,z)).$ How can I calculate the expectation $$\mathbb{E}\left[\frac{1}{b-a}\int_a^b f(t) \ dt\right]$$ ...
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Probability distribution for weights in Gym machine?

Below is the image of weights in Gym, I would like to know what probability distribution that would fit the wear and tear of the weights. Below are my initial thoughts: (1) Has to be discrete ...
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Questions around modelling arrival process of randomly sized groups

I have the following situation: I'm trying to model groups arriving to some location by some process. I assume the distribution on some interval $T$ is a Gamma-Poisson mixture where $\Lambda \sim ...
BeechAndBirch's user avatar
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Compound distribution representation

Given a distribution function, under what conditions is it possible to represent it as a mixture over Bernoulli random variables? For example, suppose you have a continuous random variable on an ...
Ralph 's user avatar
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Probability generating function and binomial coefficients

I'm reading an article where the authors derive the mass function of a compound distribution by considering the generating function. The generating function of interest for a random variable $N$ is a ...
statian's user avatar
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What is the distribution of a uniform with a bound drawn from a uniform?

Suppose I have a uniform distribution $X \sim U[a,1]$ with $a \sim U[c,1]$? How can I characterize the CDF of X?
Michael Lachanski's user avatar
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Region of significance analysis with a compound poisson linear model?

I conducted a hierarchical analysis with a compound Poisson linear model (cplm) in R, using the cplm package, and found a significant interaction effect. I want to perform a region-of-significance ...
Lior's user avatar
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Expectation of a Compound Poisson Distribution

I am trying to understand the proof of Theorem 16.14 of Probability Theory by A. Klenke (3rd version) about the Levy-Khinchin formula. I would like to know how to prove this: $$E[X]=\int x e^{-v(\...
Enrico's user avatar
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2 votes
1 answer
104 views

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\...
Mentossinho's user avatar
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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 $Y$ ...
André Goulart's user avatar
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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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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 ...
Arunthadhi Natarajan's user avatar
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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 ...
nk14's user avatar
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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 ...
guestaccount798's user avatar
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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 ...
holy_schmitt's user avatar
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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 ...
Nick's user avatar
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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 \...
PSE's user avatar
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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 ...
Golden_Ratio's user avatar
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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 \...
PSE's user avatar
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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 ...
user346481's user avatar
2 votes
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142 views

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{\...
user346481's user avatar
6 votes
0 answers
601 views

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,...
lemmykc's user avatar
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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 ...
N8_Coder's user avatar
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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 ...
Kimmel's user avatar
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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 ...
Sunny1402's user avatar
8 votes
2 answers
987 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 ...
AndreA's user avatar
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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 ...
Henrique's user avatar
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2 votes
1 answer
423 views

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 ...
Israel Barquín's user avatar
1 vote
0 answers
132 views

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 $\...
user1356855's user avatar
1 vote
0 answers
80 views

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 ...
WZhao's user avatar
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1 answer
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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 ...
SpeedyShark's user avatar
1 vote
3 answers
646 views

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 ...
masfenix's user avatar
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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))...
Migos's user avatar
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1 vote
0 answers
69 views

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{\...
Paul Harrison's user avatar
11 votes
2 answers
6k views

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 ...
Josh's user avatar
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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: $$\Bbb P(Y=k)=E_X[\Bbb P(Y=k)|X]$$
MC1325's user avatar
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1 vote
1 answer
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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 ...
AndreA's user avatar
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0 votes
2 answers
126 views

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....
Marco R's user avatar
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4 votes
2 answers
3k views

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{...
이현민's user avatar
2 votes
1 answer
68 views

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, ...
Lauren Hosking's user avatar
2 votes
2 answers
174 views

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 $Y$ = ...
Orlando's user avatar
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0 answers
146 views

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 ...
Meraou amine's user avatar
2 votes
0 answers
60 views

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 ...
joao's user avatar
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3 votes
1 answer
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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 pmf ...
pablo's user avatar
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0 answers
56 views

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, $...
Tosh's user avatar
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4 votes
1 answer
617 views

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 ...
Sextus Empiricus's user avatar
3 votes
1 answer
399 views

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}...
pablo's user avatar
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