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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What rule is used in this derivation of the interarrival time for the Poisson process?

I'm working on calculating the probability distribution of the interarrival time of the Poisson process. The method used in my textbook is very strange I don't understand how the probabilities are ...
ekke's user avatar
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Products of (spatial) non-homogenous Poisson random variables.

I have a non-homogeneous Poisson point process $X = (x_k, y_k)$ with intensity parameter $\lambda(x,y)$. For any two random points $(x_i,y_i)$ and $(x_j, y_j)$, I am forming the (product) random ...
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Necessary Assumptions when Deriving Poisson Distribution

Poisson distribution expresses the probability that a specific number of discrete independent events happen over a fixed time interval, as long as the events are sufficiently rare. To be precise, I ...
Samuel Hapak's user avatar
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Large amount of sets in Venn Diagrams and inference

Let $U$ be the universe of a multiple choice questionnaire. We can say $|U| = 1$. We have $k \geq 3$ sets of answers in $U$ such that for all $k$, we have $ S= \sum_{u_i \in U} |u_i| > 1$. The ...
Eemil Wallin's user avatar
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1 answer
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Poisson distribution with big numbers [closed]

I failed this question at an exam and would like help in explaining how to solve it: Historical data show that approximately 70 000 vehicles cross Älvsborgsbron every day. What is the probability that ...
Kristoffer Gregenäs's user avatar
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Numer of arrivals in merged Poisson Process, given number of arrivals in one process

I'm working on an exercise related to Poisson processes and have encountered a step in the solution that confuses me. The exercise is as follows: Consider two independent Poisson processes $X_1(t)$ ...
Vlad Ionescu's user avatar
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1 answer
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PMF of the sum of two independent random variables with Poisson distribution

I'm working through the problem below: Q: Let $X∼Poisson(α)$ and $Y∼Poisson(β)$ be two independent random variables. Define a new random variable as $Z=X+Y$. Find the PMF of $Z$. A: $$ \begin{aligned} ...
IGottaLearnMath's user avatar
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Power law distribution of Mixed Poisson random variable

Let $X$ a positive random variable with power law distribution with exponent $\tau$, meaning that there exists $c>0$ such that its CDF has the form \begin{equation} 1-F(x)=\mathbb{P}(X\ge x)\sim cx^...
tommycautero's user avatar
3 votes
1 answer
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Poisson's distribution for probability

This question is more for learning purposes than anything however I came across this while trying to solving the following problem: The odds of winning the lottery are 1 to 50000 million. This week, ...
d0uble_a_b4ttery's user avatar
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Derivation of Gamma distribution without using Poisson distribution

Most of the derivations of the Gamma distribution pdf I've seen on here use the Poisson distribution. My lecture notes use the Gamma distribution and the exponential inter-arrival time definition of a ...
hegash's user avatar
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Inductive proof Poisson process counts follow Poisson distribution

In converting between two definitions of a Poisson process, namely starting from the "exponential inter arrival-times" definition and attempting to prove the "Poisson distribution of ...
hegash's user avatar
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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 ...
Dan's user avatar
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Simple question on exponential distributions [closed]

Let $T$ be the time it takes for an apple to fall from a tree. Assume $T\sim \text{Exp}(\lambda)$. Now define $p(t)$ as the probability that the apple falls at time $t$, given that it has not fallen ...
sam wolfe's user avatar
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The codomain of a discrete random variable

Let $(\Omega, F, \mathbb{P})$ be a probability space and let $(S, \Sigma)$ be a measurable space. Consider a random variable $X: \Omega \rightarrow S$. We say that $X$ is discrete if its range $X(\...
mathslover's user avatar
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Why are the Binomial and Poisson probabilities different for frequency of a successful dice roll?

Why do I get different results using a Binomial vs Poisson process for calculating the frequency of a successful roll of a 6 sided die? Should I get the same answer calculating the same scenario with ...
Frank's user avatar
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Poisson probability lost sales?

"Assume that a monthly demand X is Poisson distributed with a mean of 200. Given that a grocery store can only place an order at the beginning of a month, how many item should be ordered to make ...
Fernando Martinez's user avatar
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Is any independent counting process Poisson?

I have seen a proof for the following statement: If a counting process $\left\{ N(t) \mid t\ge 0\right\}$ is homogeneous and has independent increments, i.e., $N(b_1)-N(a_1)$, $\dots$, $N(b_n)-N(a_n)$ ...
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Draw (or approximate) Poisson variable using other distributions

I'm experimenting with bootstrap resampling a large dataset, accessible via Trino SQL. One approach for resampling efficiently is drawing a variable from the Poisson distribution that gives how many ...
Bruno Kim's user avatar
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Conjecture about average cluster size in a prime-like sequence

Consider the sequence $u_n=n^a+p_n$, where $a$ is a real constant and $p_n$ is the $n$th prime. Write down the terms in $u_n$ increasing from left to right. From each term, draw a line segment ...
Dan's user avatar
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Calculate MLE only given how many non-zero RV’s.

Given RV’s $Y_1,\ldots,Y_n$ all of which i.i.d. with Poisson distribution with fixed $\lambda$. The question is to show that the MLE based on only the number of $Y_i$’s that are zero, say $n_0$, is ...
Jan's user avatar
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Prove that the following sequences monotonically decrease and increase correspondingly. Since they are bounded, find the limit.

Let $\xi_n$ be a Poisson random variable with $\lambda = n \in \mathbb{N}$. That is $P(\xi_n = k) = \frac{n^ke^{-n}}{k!}$, for $k \in \mathbb{N}_0$. Let $f_+(n) = P(\xi_n \geq n),\ f_-(n) = P(\xi_n &...
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Convergence in Probability (Law of Large Numbers)

Suppose $X_1,X_2,…,X_n$ are iid Poisson random variables, each with mean $\theta$. How to prove that $Y_n=\exp[−\frac{1}{n}(X_1+X_2+⋯+X_n)]$ converges in probability to $P(X=0)=\exp(−\theta)$ ? Hint: ...
john22445's user avatar
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Convergence in Probability (Weak Law of Large Number)

Suppose $𝑌_1, 𝑌_2, … , 𝑌_𝑛$ are independent and identically distributed Poisson random variables, each with mean $𝜆$. Prove that $𝑋𝑛 =exp[−(1/𝑛)(𝑌_1 + 𝑌_2 + ⋯ + 𝑌_𝑛)]$ converges in ...
john22445's user avatar
6 votes
2 answers
303 views

Insurance company with claims following a Poisson Process. Calculate the probability that the capital is always positive throughout the first 4 days.

Suppose that claims are made to an insurance company according to a Poisson process with rate $10$ per day. The amount of a claim is a random variable that has an exponential distribution with mean $1,...
edit_profile's user avatar
3 votes
2 answers
75 views

Given Bell numbers as moments, derive the Poisson distribution

The Poisson distribution (with $\lambda=1$) has probability mass function $\frac{e^{-1}}{k!}$ where $k\in\{0,1,2,\cdots\}$. Its moments are the Bell numbers $B_n$, which count the possible partitions ...
Andrius Kulikauskas's user avatar
1 vote
1 answer
104 views

Covariance of random variable and sum of random variables

Consider a vector $\{v_1,...,v_n\}$, $v_k\in B$, where $B$ consists of $B_1,...,B_m$ disjoint subsets and let \begin{align*} L(v_k) = \begin{cases} 1 \quad \text{, if a specific event occurs for} \...
CauchySchwarz's user avatar
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1 answer
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Inverse Poisson distribution [closed]

Maybe it's a stupid question, but I have spent hours searching answer to this with no success, so hopefully someone can help me with this. In the NBA you can bet on how many blocked shots will a ...
David Rozehnal's user avatar
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Using the Poisson distribution on two dependent events?

Problem: Suppose the number of cars passing a road follows Poisson distribution with average 5 cars per minute. Also suppose the percent of red cars follows Poisson distribution with average 20% cars ...
Vivek's user avatar
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Probability and expected value at a given time in poisson processes

Let $\{N_{t}, t \geq 0\}$ be a Poisson process of rate $\lambda=5$ and $\{T_{t}, n \geq 1\}$ its corresponding arrival times. It is given the following situations: $E(N_{10}| N_{1}=1, N_{3}=4)$ and $...
Matheus Cerqueira's user avatar
1 vote
2 answers
97 views

Variance of Poisson and Exponential distribution [closed]

I have encountered a problem with the variance of Poisson and exponential distribution. Suppose there is an accident that follows a Poisson distribution with an occurrence rate of $\lambda$ per hour. ...
Ziyao Zhang's user avatar
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2 answers
95 views

Simple question regarding the Poisson distribution

I'm just trying to wrap my head around the Poisson distribution to make it more intuitive and I'm struggling with one (seemingly) paradoxical example. Say we have a rate of 4 events in a given time ...
Daniel Podobinski's user avatar
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1 answer
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How can I show that a Poisson process with my definition below has stationary and independent increments?

We had the following definition: Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_t, \Bbb{P})$ be a filtered probability space. An $(\mathcal{F}_t)_t$ Poisson process $(N_t)_{t\geq 0}$ is a right ...
Summerday's user avatar
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2 votes
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$x$ is distributed as Poisson with parameter $\lambda$, prove $\text{E}(\log(x+0.5)-\log\lambda)+0.02/\lambda>0$

Let $x\sim\text{Pois}(\lambda)$, prove that for any $\lambda>0$, $$f(\lambda)=\text{E}(\log(x+0.5)-\log\lambda)+0.02/\lambda>0.$$ In other words, $$\sum_{x=0}^{\infty}\log(x+0.5)\frac{\lambda^x}{...
zerotrial's user avatar
1 vote
1 answer
160 views

A Poisson random variable that can't take a value of $0$

The question is as follows: The number of eggs laid on a tree leaf by an insect of a certain type is a Poisson random variable with parameter $λ$. However, such a random variable can only be observed ...
Abdo Ismail's user avatar
1 vote
0 answers
55 views

Why is the PMF of the Poisson distribution proportional to the exponential Taylor series.

Looking at the PMF of the Poisson distribution: $$P(X=x) = \frac{e^{-\lambda}\lambda^x}{x!} =e^{-\lambda} \frac{\lambda^x}{x!}$$ The $\frac{\lambda^x}{x!}$ is the $x$-th term in the Taylor expansion ...
Rohit Pandey's user avatar
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Given conjugate prior and posterior distributions, what is the PRIOR predictive distribution?

I am doing an assignment on my statistics class. We had 1 lecture about bayesian parameter estimation, where we were taught about the following formula (and it's discrete form, if $h(\theta)$ was ...
ampersander's user avatar
3 votes
1 answer
94 views

How to calculate probabilities in a Poisson process with exponential lifetime of arrivals?

I have a Poisson process where people arrive at the rate of $λ$ -- so when an event occurs, a new person arrives. This means that the times between successive arrivals are $T_i$ ~ Exponential $(\...
MilesToGo's user avatar
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1 vote
2 answers
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Valid assumptions for Poisson as limit of Binomial

Two different sources use two different sets of assumptions for the Poisson distribution as a limiting case of the binomial distribution. This video sets $$n\rightarrow\infty, p\rightarrow0, np\...
Starlight's user avatar
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Concern about "Length of Longest Run" in Statistics

My Question: In the derivation (provided below), the case where more than k consecutive heads occur isn't considered. Length of the Longest run: If a coin is flipped n times, what is the probability ...
Abhishek Kashyap's user avatar
1 vote
1 answer
81 views

Exponential Waiting times with two queues

I am trying to understand the applications of exponential waiting times when there are two queues. Let there be two counters in a mall, the first counter $X$ (where the order is placed), and the ...
Shatarupa18's user avatar
1 vote
0 answers
24 views

Finding a worst case upper bound for the Poisson mean

Assume that $X_1,\ldots X_m \sim Poi(\lambda_0)$ (iid). Now the goal is to get a worst-case upper bound for $\lambda_0$, which is also consistent when $X_1=...=X_m=0$, or more generally, when $S_m = \...
FreddyGrit's user avatar
3 votes
1 answer
101 views

If $X,Y \sim \mathscr P(\lambda)$ and $X+Y \sim \mathscr P(2\lambda)$, are $X$ and $Y$ independent?

Let $X$ and $Y$ be two random variables that follow the Poisson distribution $\mathscr P(\lambda)$ (with $\lambda > 0$). It is well-known that if $X$ and $Y$ are independent then the random ...
Héhéhé's user avatar
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1 vote
1 answer
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How can i get same result with simple Poisson distribution and the Poisson Distribution Function (CDF)?

I am comparing the sum of the results of a simple Poisson Distribution with the result of the Poisson Distribution Function (CDF)...
Zollikofen4's user avatar
4 votes
0 answers
84 views

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 ...
Andrew Joplh's user avatar
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1 answer
56 views

Does the Poisson limit theorem talk about random variables or distributions?

I am confused about the way we take a limit below when saying a Poisson is a limit of binomial, and also whether we are talking about random variables in the limit, or distributions. The textbook says ...
Princess Mia's user avatar
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Why is the Poisson distribution not necessarily symmetrical even though it has constant variance?

The variance of $X \sim$Poisson$(\lambda)$ is $\lambda$, which is a constant. Intuitively, I take this to be that the values of $X$ are symmetrically distributed; however, many Poisson distributions I ...
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For $Y \sim$Poisson$(\lambda)$, is there necessarily an $X\sim$Binomial$(n, \frac \lambda n)$ defined on $Y$'s sample space?

I have learnt that the probability distribution of a binomial random variable $X \sim$Binomial$(n, \frac \lambda n)$ converges to the Poisson distribution with parameter $\lambda$ as $n$ goes to ...
Princess Mia's user avatar
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Check proof that $\mathbb{E}[\mathrm{X}\sim \mathrm{Po}(\lambda)]=\lambda$

I am trying to show that the expectation of a Poisson variable is equal to the expected value $\lambda$. This is my proof: $$\mathbb{E}[\mathrm{X}]=\sum_{k=0}^\infty k\:\frac{\lambda^k}{k!}e^{-\lambda}...
Bosco's user avatar
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1 vote
1 answer
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Why does every interval have the same probability of success in this motivation behind Poisson Distribution?

I have been reading the following motivation behind the Poisson distribution, and have been confused why we assume that every disjoint interval has the same probability of success based on the ...
Princess Mia's user avatar
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2 votes
2 answers
93 views

Finding the mgf, expectation and variance of random sum of Poisson random variables

Question Consider the random sum $$Y = I(N>0)\sum_{n=1}^{N}X_n$$ where $\left(X_n \right)_{n\geq 1}$ is a sequence of independently and identically distributed random variables that is independent ...
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