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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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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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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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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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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…
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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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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 $X-Y=n$?
bon
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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 <…
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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 ^…
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Properties of the Minimum of Two Poisson Random Variables

I stumbled upon the following problem in my research. We are trying to analyze $Z=\min(X,Y)$ where $X \sim Pois(p\lambda)$ and $Y\sim Pois((1-p)\lambda)$. Note that the RVs expectation is related yet not identical but are independent. What we are…
eyals
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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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Why is the Poisson Approximation to the Binomial Distribution Useful?

My textbook says that if $n$ (number of trials / independent Bernoulli random variables) is very large and $p$ (probability of success per trial / Bernoulli random variable) is very close to $0$ or $1$, we can approximate the mass associated with…
user865043
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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 number of primitive roots mod $p$ that are not…
Esteban Crespi
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Let $X$ be Poisson r.v. with $\lambda$ find $f(x)$ such that $E[f(X)]=\lambda \log (\lambda)$

I am looking for a function $f(x)$ such that \begin{align} E[f(X)]=\lambda \log (\lambda) \quad \text{ for all } \lambda \ge 0 \tag{$*$} \end{align} where $X$ is a Poisson random varaible with parameter $\lambda$. Note, we are looking for a…
Boby
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Closed form of series with factorial-squared denominator?

Does the following series have a closed-form expression: $$\sum_{k=0}^{\infty} \frac{z^k}{(k!)^2}$$ I know that it must converge because: $$\sum_{k=0}^{\infty} \frac{z^k}{k!} = e^z$$ and the $(k!)^2$ denominator obviously increases more quickly than…
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