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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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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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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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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Is there a closed-form expression for the Poisson Lognormal pmf?

Suppose the $\lambda$ parameter of a Poisson distribution is generated from a $LogNormal(\mu, \sigma)$ distribution. Can the final pmf be expressed with only elementary functions? $$f(x;\mu,\sigma)=\...
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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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150 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 ...
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MLE estimation of poisson intensity for a normal compounded poisson process

I was working on a project where a time series is modeled as compound poisson distribution with normally distributed jump levels and I want to get MLE estimator for the poisson intensity, jump mean, ...
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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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86 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}...
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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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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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45 views

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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969 views

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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220 views

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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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 ...
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Compound distribution in Bayesian sense vs. compound distribution as random sum

I'm trying to sort out two different uses of the term "compound distribution" and figure out the relationship. The Wikipedia article on compound distribution -- which I wrote -- defines a compound ...
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206 views

What is a good reference for compound Poisson processes?

I've seen a couple of descriptions of the basic statistics of a compound Poisson process, basically just simple statements about how to compute the mean and variance given the mean and variance of the ...