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.
66
questions
0
votes
0
answers
25
views
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 ...
- 21
6
votes
1
answer
252
views
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 ...
- 429
0
votes
0
answers
26
views
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?
0
votes
0
answers
8
views
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 ...
- 41
1
vote
0
answers
41
views
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:
$$E[X]=\int x e^{-v(\...
- 211
2
votes
1
answer
82
views
Compound binomial distribution distributed as binomial
Suppose we have independent family of random variables $\{Y\}_{i\in\mathbb{N}}\cup\{N\}$, where $Y$s are identically distributed.
Next consider a sum of random number of random variables $W_N\equiv\...
- 342
0
votes
0
answers
113
views
A question about the skewness of a compound Poisson distribution
I'm trying to intuitively answer the last part of item b of question 21 of chapter 9 (Mixtures and Compound Distributions) of the book Probability: the science of uncertainty with applications to ...
- 997
0
votes
0
answers
30
views
Which is $E[L]$ and $Var(L)$ in Compound Poisson (in terms of F function)?
I have a compound Poisson model that calculates the danger to my business. I know that the loss to my business in a time interval $T$ is $L = \displaystyle\sum_{i=1} ^{N(T)} X_i$ with $N(T) \sim Poi(\...
- 123
0
votes
0
answers
95
views
Is every Compound Poisson distribution a Mixture model?
We have two models:
Let $N \sim \hbox Poisson (\lambda)$ and let $(X_k ; k =1,2,3,...)$ be a a sequence of independent and identically distributed random objects (random variables, vectors or ...
- 997
0
votes
0
answers
62
views
An estimation problem related to a certain stochastic process (radom sum of stochastic processes)
Let $N\sim \hbox{Poisson}(\lambda)$ and $(X_t)_{t \in \mathbb Z}$ be a stochastic process. Consider the following stochastic process:
$$ Y_t = \sum_{j=1}^N X_{t,j}$$
where $(X_{t,1})_{t\in \mathbb Z}, ...
- 997
2
votes
1
answer
238
views
A question about conditional expectation
Let $X_1,X_2,...,X_n$ and $Y$ be random variables. I know that:
$$\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] $$
Now, suppose that $Y$ ...
- 247
2
votes
1
answer
114
views
Trying to understand if a model is an empirical Bayes or not
I am trying to understand if the model published in section 4.1.1 here is or is not an Empirical Bayes model (which the author claims it is). Or, maybe, if it is a valid one or not. The model looks as ...
- 157
0
votes
0
answers
70
views
How can compound a exponential distribution with gamma distribution?
A random variable X is said to have a exponential distribution with parameter lamda > 0. if its pdf is given by f(x,lamda)=(lamda * exp(-lamda * x)).
A random variable X is said to have a gamma ...
0
votes
0
answers
86
views
Binomial distribution conditional on previous sampling
When using a Binomial distribution, the bulk of literature assumes that the probability of success $p$ is known, or deterministic.
I would like to get the distribution for a Binomial RV with unknown ...
- 1
0
votes
1
answer
60
views
Compounding Gamma with Gamma to yield F-distribution?
I am working through some problems from my Bayesian Statistics course and am having trouble understanding a step in the solution to a question. For reference this is the question:
And here is the ...
0
votes
0
answers
87
views
How does posterior predictive mean depend on parameters of the likelihood and prior distribution?
I have come across a problem in my research which deals with the mean of the posterior predictive distribution, i.e.
$$p(x'|x)=\int d\theta p(x'|\theta)p(\theta|x)$$
where $x$ is an observed sample ...
- 111
2
votes
0
answers
55
views
How to specify a parameters for Gamma distribution?
I have a task: A frequency claim distribution, $K$ is a compound Poisson-Gamma distribution. The mean of the Poisson distribution is gamma distributed with mean equal to 1 and variation equal 2. Find ...
- 848
1
vote
0
answers
109
views
Find the characteristic function of a random variable involving a compoud poisson distribution
Let $\{\xi_i \}_{i=1}^{k}$ be a finite sequence of independent r.v. such that $\xi_{i} \sim F_i$.
For each $i$, let:
$\{\xi_{i,j}\}_{j=1}^{\infty}$ be a sequence of copies of $\xi_i$, that is: $\xi_{...
- 997
3
votes
0
answers
153
views
What is the distribution of the Poisson Sum of gaussians?
I know that the sum of two independent normal random variables is normal. Particulary, when one is copy of other, i.e., if $X_1, X_2 \sim \mathcal{N}(0,\sigma^2)$, independent, we have:
$$X_1 + X_2 \...
- 256
1
vote
0
answers
47
views
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 ...
- 533
1
vote
0
answers
133
views
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 \...
- 256
3
votes
0
answers
137
views
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 ...
- 51
2
votes
0
answers
124
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{\...
- 51
6
votes
0
answers
511
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,...
- 91
2
votes
1
answer
167
views
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 ...
- 189
1
vote
0
answers
19
views
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 ...
- 11
0
votes
1
answer
34
views
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 ...
8
votes
2
answers
857
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 ...
- 225
1
vote
0
answers
35
views
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 ...
- 167
2
votes
1
answer
368
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 ...
1
vote
0
answers
128
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 $\...
- 111
1
vote
0
answers
74
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 ...
- 111
0
votes
1
answer
831
views
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 ...
- 11
1
vote
3
answers
541
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
$$
z \mid y \sim Binomial(\pi, y) \\
y \sim ...
- 481
4
votes
3
answers
1k
views
Poisson Distribution with Exponential Parameter
If we have $X(k)\sim Pois(2k)$ and $Y \sim Exp(15)$ and $Z=X(5Y)$. How can we determine $E(Z)$, $Var(Z)$ and $P(Z = z)$.
So far I'm thinking
$$\begin{align*}
E(Z) &= E(X(5Y)) \\
&= E(Pois(10Y))...
- 142
1
vote
0
answers
67
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:
$$
S \sim \chi^2_{\nu_2} / \nu_2
$$
$$
F \sim \frac{\...
- 391
11
votes
2
answers
5k
views
Poisson Gamma Mixture = Negative Binomially Distributed?
This paper introduces a model called "Beta-Geometric / NBD" which models "repeat-buying behavior in settings where customer “dropout” is unobserved: It assumes that customers buy at a ...
- 4,008
1
vote
1
answer
146
views
finding PDF of Y, given Y|X [closed]
$$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]$$
- 43
1
vote
1
answer
177
views
Stratified sampling / QMC simulation for compound Poisson rv
I have a rv $X$ of the form
$$ X=\sum_{i=1}^N Y_i, $$
where $N$ is a discrete rv (often, but not always, Poisson) and $Y_1,\ldots,Y_N$ are continuous random variables, iid and independent from $N$.
I ...
- 225
0
votes
2
answers
119
views
Parameter estimation for random variables where a control parameter is another r.v
Let $\{X_i\}$ a sequence of independent random variables.
Each $X_i$ has a p.d.f $p(m, \theta)$. Where $\theta$ is a real unknown parameter and $m$ the outcome of another random variable $M$ with p.d....
- 63
4
votes
2
answers
2k
views
What is the distribution of a mixture of exponential distributions whose rate parameters follow a gamma distribution?
I want to know the theoretical distribution of a mixture of exponential distributions whose rate parameters are distributed according to a gamma distribution:
$$ y\sim\text{Exp}(\theta), \quad\text{...
- 41
2
votes
1
answer
66
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, ...
2
votes
2
answers
172
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 $Y$ = ...
- 61
0
votes
0
answers
140
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 ...
2
votes
0
answers
55
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 ...
- 21
3
votes
1
answer
1k
views
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 ...
- 165
0
votes
0
answers
51
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, $...
- 101
4
votes
1
answer
563
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 ...
- 71.9k
3
votes
1
answer
380
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}...
- 165
2
votes
0
answers
14
views
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 ...
- 3,047