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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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What is the relationship between poisson, gamma, and exponential distribution?

I'm having a hard time understanding the intuitive relationship between these three distributions. I thought that poisson is what you get when you sum n number of exponentially distributed variables, but if seems that gamma is the same...Could…
xyy
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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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Poisson distribution with an integer $\lambda$ value

I have noticed that when a Poisson distribution has an integer value of $\lambda$, the following holds: $$ \mathbb{P}[X = \lambda] = \mathbb{P}[X = \lambda - 1] $$ I have been able to prove this rather simply using an algebraic method. However, my…
willmaths
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Intuitive explanation of Poisson distribution

I've seen the formula most commonly derived as a continuum generalization of a binomial random variable with large $n$, small $p$ and finite $\lambda = np$ yielding $$ \lim_{n \to \infty} \binom{n}{x} p^x(1-p)^{n-x} = e^{-\lambda}\frac{\lambda ^…
jaslibra
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How to derive the likelihood and loglikelihood of the poisson distribution

As the title suggests, I'm really struggling to derive the likelihood function of the poisson distribution (mostly down to the fact I'm having a hard time understanding the concept of likelihood at all). I've watched a couple videos and understand…
user416936
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43 cookies are randomly given to 10 children. What's the probability each child receives at least 2 cookies?

I wanted to ask 1) if I've solved this puzzle problem correctly, and 2) if there is a shorter or more elegant approach. There are 43 cookies to be given out at random to 10 children. What is the probability that each child gets at least 2…
ctesta01
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sum and difference between two independent Poisson random variables

Let $X$ and $Y$ be independent Poisson variables with respective parameters $a$ and $b$. What is the distribution function of $X+Y$? the conditional distribution of $X$, given $X+Y=n$? same, given $X-Y=n$?
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Showing $E(S^2\mid \bar X)=\bar X$ for i.i.d Poisson random variables $X_i$

Let $X_1,X_2,\ldots,X_n$ be i.i.d $\text{P}(\lambda)$ random variables where $\lambda(>0)$ is unknown. Define $$\bar X=\frac{1}{n}\sum_{i=1}^n X_i\qquad,\qquad S^2=\frac{1}{n-1}\sum_{i=1}^n(X_i-\bar X)^2$$ as the sample mean and sample variance…
StubbornAtom
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What is the probability of getting an even number from a Poisson random draw?

Below is a graph showing the probability of drawing an odd number (y-axis) from a Poisson distribution with a given expected value (x-axis) x = seq(0,1e4,1) // range of values to explore lambdas = seq(0,4,0.01) // expected value of the…
Remi.b
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Expected Value of Square Root of Poisson Random Variable

Find the expected value of $\sqrt{K}$ where $K$ is a random variable according to Poisson distribution with parameter $\lambda$. I don't know how to calculate the following sum: $E[\sqrt{K}]= e^{-\lambda} \sum_{k=0}^{\infty} \sqrt{k}…
Susan_Math123
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What is the difference between a Poisson and an Exponential distribution?

For a Poisson distribution: $$\mathsf{P}(X=x)=\frac{e^{-\mu}\times \mu^x}{x!}$$ where $\mu$ is the mean number of occurrences. For an Exponential distribution: $$f(x;\lambda) = \begin{cases} \lambda e^{-\lambda x} & x \ge 0 \\ 0 & x <…
BLAZE
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A question about Poisson processes

Reading through the book "Brownian Motion & Stochastic Calculus" by Karatzas and Shreve, I found the following exercise (problem 3.9, page 15): Let $ \ N \ $ be a Poisson process with intensity $ \lambda > 0 $ (this means, in particular, that $ N_t…
Mad Si
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How to prove Poisson Distribution is the approximation of Binomial Distribution?

I was reading Introduction to Probability Models 11th Edition and saw this proof of why Poisson Distribution is the approximation of Binomial Distribution when n is large and p is small: An important property of the Poisson random variable is that…
Demonedge
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Expected maximum of a sequence of i.i.d. Poissons

Let $X_i \sim \mathrm{Pois}(1)$ be a sequence of $n$ i.i.d. random variables (with Poisson distribution with parameter 1). I'm interested in the asymptotic behavior of $$\mathbb E[\max_{i \in \{1\ldots n\}}X_i],$$ i.e., the expected maximum value of…
Alf
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A walk on a $2D$ Poisson process in which every step goes to the nearest unvisited point: expected distance from origin after $365$ steps?

Uncle's epic journey One year ago, my uncle set off from our village on an epic journey, in which every day he travels to the nearest unvisited village and stays there for the night. The villages in our country are randomly located (independently…
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