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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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How can Poisson distribution predict probability with so little information?

Forgive my ignorance, I am brand new to Poisson and statistics in general. $$ \bbox[5px,background:black]{\color{white}{\begin{array}{l} \text{Poisson Distribution}\\ P(X=k)=\frac{\lambda^ke^{-\lambda}}{k!}\\ k\text{ is the given number of event…
Ryan
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Does Benford’s law hold for the Poisson distribution?

According to the Wikipedia page of Benford’s Law: Benford's law was empirically tested against the numbers (up to the 10th digit) generated by a number of important distributions, including the uniform distribution, the exponential distribution,…
Riemann
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deriving mean & variance for poisson using mgf

I'm trying to derive the mean and variance for the Poisson distribution but I'm encountering a problem and I believe its due to my derivatives. So the mgf for poisson is: $M_x(t)=e^{\lambda e^t-\lambda}$ where $x=0,1,2..$ and $\lambda \geq 0 $ So to…
Caddy Heron
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Poisson Distribution of Underfilled Bottles

3 bottles per case are underfilled on average. What is the chance that at least 4 underfilled bottles will be contained in a random case? Using the formula: $1-\left( \dfrac{3^0 e^{-3} }{ 0!} + \dfrac{3^1 e^{-3}}{1!} + \dfrac{3^2 e^{-3}}{2!} +…
Florencio
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How do I find $\lambda$ in the poisson distribution?

So I am a bit confused. Say $1$ person in $1000$ forget to clean their hands after going to washroom. If $10,000$ go to the washroom, what is the probability of $6$ people forgetting to wash their hands? So, from my knowledge, I would assume that…
Belphegor
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Poisson distribution for accident rate and bonus to be given

Here is the problem. A car factory has an accident rate of 0.3 per month. a) Find after how many months we expect at least one accident with a probability more than 85%. b) To motivate its workers, the company pays the union a bonus of 5000USD if…
Just_Newbie
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Distribution of difference between independent Poisson random variables

I'm working on a problem in which the random variable is the difference between two random variables. We know from math that the mean will be the difference of the means, and that the variance will be the sum of the variances. I know that for a…
gciriani
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Poisson distribution?Probability of expected number occuring?

Cars pass through a road junction according to Poisson distribution.An average of 7 cars per minute pass through this junction. What is the expected number of cars passing through in 3 minutes? My answer is $3 \times 7=21$ What is the probability…
Sook Lim
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how to solve this Poisson distribution problem?

I got this Poisson distribution problem. There's a clothes shop, and average number of customers per 1 hour are 5 men , 10 women, independent each other, in Poisson distribution. If for 30 minutes there had been totally 10 customers, what's the…
서영빈
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Expectation of $\bar{Y}^3$

How do I calculate $\mathbb{E}\left[\bar{Y}^3\right]$? $Y$ has a Poisson distribution with mean $\lambda$ , and there are $n$ independent $Y$'s. Do I find $\mathbb{E}\left[Y^3\right]$ first? How do I go from there then?
L.mak
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Scaling a Poisson distribution

Is it possible to scale a Poisson distribution and receive the same result. Lets say that I have bridge A. On average 10 cars drive over bridge A per hour, thus if I want to calculate the probability that at most 4 cars drove over bridge A after a…
no nein
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poisson distribution find exact sum

Using only properties of Poisson distribution find the exact value of this sum $\sum_{x=0}^{\infty}\frac{x^22^x}{x!}$ I believe $\lambda$ = 2 (E(X))$^2$ = 4 V(X) = 2 E(X$^2$) = 6 I don't know how to find the exact value of the sum.
user284064
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poisson process modelling

Consider a birth process, where at each unit time step $t$, on the average, there is a fraction of $0.012$ among all the populations who will give a birth. Suppose the doubling time is $m$, then, from $1.012^m =2$ we get that $m$ is approximately…
nstrong
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Probability of two players scoring in a soccer match (Poisson)

Assume that the score in a soccer match follows a Poisson distribution, with Team A expected to score 1.35 goals in a particular game (lambda = 1.35). Given player X is expected to score 33% of Team A's goals and player Y is expected to score 25%,…
clattenburg cake
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$E[X\log X] - \lambda \log \lambda$ when $X \sim Poisson(\lambda)$ as $\lambda \rightarrow \infty$

I know the answer must be $\frac{1}{2}$, because I have done the numerical simulation and as the parameter of the poisson distribution increases, the quantity gets closer to 0.5. Can we show this analytically? Edit: For whoever comes across this in…
Eaman
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