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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.

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Question on Poisson Process

I am trying to solve the following problem: Jhon is a newspaper vendor. His customers arrive according to a Poisson process with a rate of one per minute. At any given moment, what is the probability ...
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generalized poisson moment function approach to initalize input output model

I am trying to understand how the Generalized Poisson Moment Functions apporach is a way to initalize the nominator and denominaor of a LTI-Model that describes the connection between Input and Output ...
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How can I calculate the rate of overlapping events between two Poisson processes?

I have the following problem: I have 2 independent Poisson processes with a known rates λ1 and λ2. The events in each process have a well-defined duration, D1 and D2. From that, I need to know: What ...
user2934303's user avatar
1 vote
1 answer
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Poisson Question - number of signals received at station A

the question: ...
CallMeDave's user avatar
4 votes
1 answer
231 views

Probability of two independent random variables are equal decreases when they are more spread out

Let $X_{1},X_2\sim Bin(n,p)$ be iid binomial random variables. Then it is intuitive that $\mathbb P(X_1=X_2)$ decreases when $n$ increases (keeping $n$ fixes). However, it is not obvious to prove this ...
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Distribution of binned Poissonian variable

I'm investigating a crude model of shot noise in camera pixels. Mathematically, I'm interested in the distribution of a random variable $\lfloor n/C\rfloor$, where $n \sim \text{Pois}(\lambda)$ and $...
Yly's user avatar
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2 votes
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$24$ meteorites reach the Earth's surface each year. Prob. that no more than $3$ such meteorites will come in one month. Solution verification.

An average of $24$ meteorites of a fixed size reach the Earth's surface each year. Determine the approximate probability that no more than 3 such meteorites will arrive in September 2028. Given that ...
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One journalist makes $2$ typos per article, second $3.2$ and third $4$. Prob. that a random article contains at least 2 typos. Solution verification.

The newspaper employs three journalists: A, B and C. A makes an average of $2$ typos per article, for B and C the averages are $3.2$ and $4$, respectively. The Monday edition of the newspaper contains ...
thefool's user avatar
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2 answers
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Is $\frac{1}{n}\sum_{i=1}^{n}X_i$ is a sufficient estimator for $\lambda$ in the Poisson distribution?

I know from this question that $\sum_{i=1}^{n}X_i$ is a sufficient estimator for $\lambda$ in the Poisson distribution. However, from looking at the proof I can see that $\frac{1}{n}\sum_{i=1}^{n}X_i$ ...
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Compound Poisson Distribution/Mixture of Distributions

If for example a driver causes N accidents in a given year, and N has a Poisson Distribution with parameter λ. That λ parameter itself has a uniform distribution on an (0,1) interval. Is this a ...
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UMVU estimator of $\lambda^2$ via Rao-Blackwell

I have been working on a problem, which goes as follows: Given the statistical model $(\mathcal{X},\mathcal{B},\mathcal{P})$, where $\mathcal{P}=\{P_{\lambda}^{\otimes}:P_{\lambda}=Pos(\lambda), \...
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Expectation of a Poisson random variable

Let $N_{\lambda}$ be a random variable having Poisson distribution with probability mass function given by $$ P( N_{\lambda} = k) = \frac { \lambda^k e^{- \lambda} }{ k!}, $$ where $\lambda >0.$ ...
Rye's user avatar
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UMVUE of $\mathbb{P}[X=x_0]$ where $X_1,\cdots,X_n$ is Poisson

I came up with this problem and tried to solve it myself. Please check my solution, I am a bit unsure it is correct because it says the UMVUE for the probability where $x_0>\text{sum of ...
harrydiv321's user avatar
2 votes
1 answer
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Average number of calls coming into office X, in one hour, is $4$, to Y is $7$. What is the prob. that on some day $100$ calls will come to both?

A certain company has two offices, $X$ and $Y$, operating independently, $8$ hours a day. The average number of calls coming into office X, in one hour, is $4$. The average number of calls coming into ...
thefool's user avatar
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Approximating the poisson distribution using normal distribution

The number of calls $X$ to a telephone exchange during the busiest hour of the day follows a Poisson distribution $Po(λ)$. Over $8$ days, the following independent observations of $X$ have been ...
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Turning Nearest Neighbour Distribution of Poisson Scatter Theorem to Rayleigh Distribution By Multiplying Constant

Consider a Poisson random scatter of points in a plane with mean intensity $\phi$ per unit area. Let R be the distance from 0 to closest point of the scatter. Show that $\sqrt{2\phi\pi}$R has the ...
BurgerMan's user avatar
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Rick durrett Stochastic process - Theorem 2.12

Following is a theorem statement in Rick Durrett's book on Stochastic processes. Theorem 2.12. Suppose that in a Poisson process with rate A , we keep a point that ...
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1 answer
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Determine the distribution of $X$

Let $N$ a random variable such that $N \sim \operatorname{Pois}(\lambda)$. Furthermore, let $X$ a random variable such that $$\mathbb{P}( X = k\ | \ N=n)=\binom{n}{k}p^kq^{n-k},\ \ 0\leq k \leq n, \ \ ...
Nicolas Rodriguez's user avatar
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3 answers
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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 ...
ctesta01's user avatar
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Sum of $Y$ i.i.d Binomial Random variables $(n,p)$, where $Y$ follows poisson distribution

The Question Let $X \sim \operatorname{Bin}(n, p)$, and $Y \sim \operatorname{Poisson}(\lambda)$. Let $$ T=X_1+X_2+\cdots+X_Y, $$ with $X_i{ }^{\prime}$ 's i. i. d. $\operatorname{Bin}(n, p)$ (and ...
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Derivation of Expected Number of Occurrences in a stochastic Intensity Poisson Process Using Given Axioms

I have a question concerning the expected number of occurrences in a random intensity Poisson process during a specific interval. Let $N_t(h)$ be a random variable counting the number of events that ...
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Find the probability that strictly $2$ people arrived in the first hour in a Poisson Process

Let the number of people arriving at a shop within the time interval $[0,t]$ be $X_t$. $4$ customers arrived in the first $2$ hours. Find the probability that strictly $2$ people arrived in the first ...
Rory-Laughlin's user avatar
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1 answer
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Help me intuit Equal Probabilities in Poisson Distribution for $k = λ$ and $k = λ-1$

I was trying to understand Poisson distribution and I'm confused as to why the likelihood of $k=λ-1$ is equal to $k=λ$. Here is my understanding and where my confusion is: For a given time interval, ...
Tyler Short's user avatar
1 vote
1 answer
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UMVUE of $\mathbb{E}[X^2]=\lambda^2 + \lambda$ where $X\sim\mathrm{Pois}(\lambda)$.

This is the same question as this: UMVUE of $E[X^2]$ where $X_i$ is Poisson $(\lambda)$. Here, I restate the problem for completeness: Let $X_1, \ldots, X_n \overset{\text{i.i.d.}}{\sim} \mathrm{Pois}...
pbb's user avatar
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4 votes
1 answer
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Finding the limiting distribution of $T_{n}/S_{n}$ as n tends to infinity

Question Let $X_i \sim\left(i . i\right.$. $d$.) Bernoulli $\left(\frac{\lambda}{n}\right), n \geq \lambda \geq 0$. $Y_i \sim\left(i\right.$ i. d.) Poisson $\left(\frac{\lambda}{n}\right),\left\{X_i\...
Debu's user avatar
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0 answers
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How to analyze a vector with its entries following independent distributions?

Suppose let's say we have a vector $Y$ which is distributed as $Poisson(X)$, where $X$ is an $N \times 1$ vector. Does it mean that the individual elements in $Y$ are Poisson distributed with the mean ...
Aravind Muraleedharan's user avatar
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Poisson log-likelihood parameter estimation with uncertainty on model rate

I have data $k_n$ that I expect to follow a Poisson distribution $P_{\lambda}(k)=\frac{\lambda^k e^{-\lambda}}{k!}$, and a model for the event rate $\lambda$ that depends both on a vector of unknown ...
Cullen Abelson's user avatar
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0 answers
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Poisson distribution coverage probability for confidence interval.

I have been trying to solve this problem 9.24 of Statitical inference by Casella and Berger. I was able to understand in the theory that confidence interval for Poisson distribution is based on chi-...
Mohit Sharma's user avatar
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1 answer
15 views

Poisson distribution inequality for sum of rates

Given $s,t \in \mathbb{R}^+$ and $i,j \in \mathbb{N}$, Let $X,Y,Z$ be random variables with Poisson distribution with rate $s,t$ and $s+t$, respectively. Is it true that $$ P(Z = i+j) \geq P(X=i) \...
John Kevin's user avatar
1 vote
1 answer
87 views

Expected number of passengers in a bus, if both bus and passengers arrival time have a Poisson distribution.

Here's the full question In Poisson Bus City, there is a shuttle bus that goes between Stop A and Stop B, with no stops in between. The times at which the bus arrives at Stop A are a Poisson point ...
fresh_start's user avatar
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0 answers
41 views

Why does the poisson regression beta coefficient ML estimate have no closed form solution?

Every source I have read says that there is no closed form solution for the beta coefficient but I have not seen an explanation as to why. I tried to solve for the beta coefficient on my own to see ...
decapicone's user avatar
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1 answer
49 views

Why do we use a Poisson distribution here rather than Binomial?

Approximately 80,000 marriages took place in the state of New York last year. Estimate the probability that for at least one of these couples, (a) both partners were born on April 30 I understand how ...
Mitchell's user avatar
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1 answer
23 views

What's the optimal way to approximate a binomial distribution with a Poisson distribution?

Conventionally, we approximate a binomially-distributed variable $X\sim B(n, p)$ with the Poisson-distributed variable $Y\sim Po(np)$, with the mean of $X$ and $Y$ being identical. However, we could ...
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Number of distinct urns given randomly selected balls

Suppose, we have N distinct urns each having some number of balls $\ge$ 1. The distribution of balls in the urns follows a zero-truncated Poisson distribution with given $\lambda$. If we take out all ...
aishik roy chaudhury's user avatar
1 vote
0 answers
34 views

Is my construction almost-surely a conditional Poisson random variable?

Suppose a continuous random variable $$L \sim F_{L}$$ where $$Pr[L = 0] = 0$$ even though $$L = 0$$ exists for some countable subset of the outcome space. I am considering the existence of a variable $...
Galen's user avatar
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0 votes
1 answer
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Proving the general formula for conditional expectation of a Poisson Process

I am studying a course on Stochastic Processes and encountered the following proof exercise on Poisson Processes: If $N$ is a Poisson Process with intensity $\lambda$, then for $0<s<t$ where $k ...
FD_bfa's user avatar
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1 vote
2 answers
77 views

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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0 answers
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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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1 answer
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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 ...
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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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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
47 views

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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34 views

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
64 views

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, ...
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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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9 votes
1 answer
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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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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(\...
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2 votes
1 answer
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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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1 vote
1 answer
118 views

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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