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Questions tagged [poisson-distribution]

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

The PMF of a random variable $X$ distributed according to the Poisson distribution with parameter $\lambda > 0$ is the following: $$\Pr\left[X=k\right]=\frac{\lambda^k \exp(-\lambda)}{k!}\;,\; k\geq 0$$ This distribution describes the number of independent events occurring with constant rate in some unit time, the average being $\lambda$ events per unit.

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Poisson random variable on uniformly distributed variables

Let $U_1, U_2, ...$ be $i.i.d$ random variables that are uniformly distributed on $(0,1)$. We define: $N((0,s]) = sup(k \in N: \sum_{i=1}^k (-log(1- U_i)) \leq s)$ Is $N((0,s])$ a Poisson variable with parameter $s$ ?
BlueFx
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Poisson distribution - find value for $\lambda$ given a known probability

Particles in a radioactive piece of material are decaying, and it is known that the number of decayed particles during a time period t (sec) is $Po(\lambda t)$-distributed. It is also known that the probability of there being at least one particle…
Eiraus
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Poisson distribution: P[-infinity]

I am confused with how to handle the probabilty of $P[-\infty]$ Let X be a random variable with Poisson probability distribution and $\lambda = 1.$ What is the probability $P(X>1)$? $P(X > 1) = 1 - P(X < 1)$ $P(X < 1) = P[0] + P[1] + P[-\infty]$…
user3067059
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Almost sure limit of $X_n/n$, for $X\sim Poi(n)$, Chernoff bound

I need the following result in my bachelor's thesis, and want to better understand the proof of it. $X_{n}$ independent and $X_n \sim \mathcal{P}(n) $ meaning that $X_{n}$ has Poisson distributions with parameter $n$. What is the $\lim\limits_{n\to…
blubby
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Showing conditional poisson is binomial distributed

For a homogeneous Poisson process N on [0,$\infty$] show that for 0 $$P(N(s)=k \ | \ N(t))= {{N(t)}\choose{k}} (\frac{s}{t})^k (1-\frac{s}{t})^{N(t)-k} $$ for $k\leq N(t)$ N(t) is Poisson distributed as in the Cramér-lundberg model. Is first…
Viktor Jeppesen
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Mean Deviation about mean for Poisson Distribution?

Please help on how would we solve the summation while deriving mean deviation about mean for Poisson Distribution.
Rajat Rautela
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Poisson distribution, showing the mode of $X$

I need help solving part b of this question: The random variable $X \sim ~Po(m)$. Given that $P(X=k+1)=P(X=k-1)$, where $k$ is a positive integer, a) show that $m^2=k(k+1)$ b) hence show that the mode of $X$ is $k$.
The 'Fe' Man
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A Poisson Question

IB Poisson Question A ferry carries cars across a river. There is a fixed time of $T$ minutes between crossings. The arrival of cars at the crossing can be assumed to follow a Poisson distribution with a mean of one car every four minutes. Let…
Mr T
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2 Poisson Distribution Questions

Given that the number of emergency admission to a hospital each day has a Poisson distribution with mean 2. 1) The hospital has 4 bed for emergencies at the beginning of each day. Calculate the probability that the number is insufficient for that…
Mun Yee
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Trying to understand Poisson Distribution

I'm trying to understand Poisson Distribution but I can't find a good explanation for one of the sentences above. I've highlighted it within a red rectangle - "The probability of an event in a small interval is proportional to the length of the…
glendon
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Probability Binomial Inter-arrival Time

I think I've solved part a, but would like confirmation, not 100% sure. For part b, I'm pretty lost. I think it's a poisson distribution because it's modeling wait time? But at the same time, it has frames between arrivals, so I'm not sure. Let…
Asif Sheikh
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How to use Poisson Table

The Question is here A life insurance salesman sells on the average 3 life insurance policies per week. Use Poisson's law Pr ( = ) = −33 / ! to calculate the probability that in a given week he will sell 2 or more policies but less than 5…
laish138
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Solving a Poisson Probability question

I need help in understanding what is meant by this question. Given that: λ = 0.2 incidents per week X = incidents that occur in a 52-week period. Given that no incidents occur in the first 10 weeks, find the probability that at most 10 incidents…
Zealox
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Calculating Poisson process problem

A shop sells both hot and cold drinks. Hot drink sales occur at the instants of a Poisson process with expectation 30 drinks per hour.Cold drink sales occur at the instants of a Poisson process with expectation 20 drinks per hour. 60% of customers…
whoisit
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splitting of poisson variables

this is some lecture slides from my school. I don't understand why we need to sum up all from $m=0$ to $m=+ \infty$. I think $m$ is a fixed number which represents the number of type $2$ event? plz help me! many thanks in advance!
whoisit
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