# Questions tagged [poisson-distribution]

For questions relating to Poisson distributions in probability theory. To be used with [probability] or [probability-distributions] tag.

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### Number of primitive roots mod $p$ that are not primitive roots mod $p^2$

Consider the primitive roots of a prime $p$ in the range $1...p$ which are not primitive roots mod $p^2$. Let $n(p)$ be this number. While looking for an answer to this question, it seems that the ...
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### Show that $\frac1n\log X_n$ converges almost surely

Let $X_0$ follow $\mathrm{Uniform}(0,1)$. Define $X_{n+1}$ iteratively as $X_{n+1}$ follows $\mathrm{Uniform}(0,X_n)$, $n\geq0$. Show that $\dfrac{\log X_n}{n}$ converges almost surely and find the ...
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### A weighted sum of independent Poisson random variables $X_1 + 2X_2 + 3X_3+\dots+nX_n$

I have that for $1 \leq i \leq n$, the mutually independent random variables $$X_i \sim \text{Poisson}(\mu_i)$$ Then what is the distribution of $$Y \sim \sum_{i=1}^{n}i X_i$$ It looks a bit like an ...
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### P.G.F. of total progeny in a Poisson branching process

Let $c<1$. Let $X$ be a random variable with distribution: $$\forall k\in\mathbb{N}:\Pr[X=k]=\frac{e^{-ck}\cdot (ck)^{k-1}}{k!}$$ In fact, $X$ is an r.v. describing the total progeny in a Poisson ...
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### When to use Poisson distribution?

I'm still very confused regarding when to use probability distributions. For instance, this is the assumptions to use Poisson distribution, according to Wikipedia: k is the number of times an event ...
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### Find a compund Poisson variable with characterist function as centered compound Poisson

I know that if $(X_j , j \geq 1)$ is a sequence of i.i.d. process with $\sigma$ being the probability distribution of $X_j's$ and $N \sim \text{Poisson}(\lambda)$ independent of the $X_j's$, then the ...
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### Expected value of falling factorials from axioms of Poisson process

Falling factorial, $(x)_n$, is the product of biggest $n$ terms in factorial, $(x)_n = x(x-1)(x-2)\cdot \ldots \cdot (x-n+1)$. Or the number of ways to color the set of $n$ objects into different ...
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### Distribution of no. of siblings of a random child if the no. of children of a family is Poisson distributed

Consider a large population of families, and suppose that the number of children in the different families are independent Poisson random variables with mean $\lambda$. Show that the number of ...
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### Law of large numbers; Poisson distribution

Let $X_n$ be the numbers of job applications at a company in the year $1900+n,n\in\mathbb N$. Let $(X_n)_{n\in\mathbb N}$ be a sequence of independent, identically distributed random variables with ...
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### Application Problem: Conditioning Poisson Process

I am trying to solve the following application problem: There are $n$ components with independent lifetimes which are such that component $i$ functions for an exponential time with rate $\lambda_i$. ...
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### What is the pdf of $X$, where $dX_t = -aX_t + d N_t, N_t$ is a compound Poisson process?

I would like to find the probability density function (at stationarity) of the random variable $X_t$, where (I'm not sure this notation makes sense, I'm not very familiar with the stochastic calculus ...
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### Probability of a egg surviving with a Poisson distribution

The probability that a bird deposits $r$ eggs in its nest is given by the Poisson distribution with parameter $\lambda$. Assume that the probability of a egg to survive in nature is $p$ and that the ...
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### Poisson distribution for calculation of number of calls

A roadside assistance center receives 2 calls every half hour on average. a) Find the probability for this center to get at least 3 calls in two hours. b) If the center received 3 calls in the first ...
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