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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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$U_1,U_2,…$ i.i.d. $U[0,1]$, $P\sim Poi(\lambda)$, find $F_{\operatorname{min}(U_1,…,U_P)}$

Let $(U_n)_n$ a sequence of random variables i.i.d $U[0,1]$ and let $P\sim Poi(\lambda)$ a random variable such that $P$ is independent of $(U_n)_n$. Let $$ \\ X=\left\{\begin{matrix} \operatorname{...
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Conditional probability of Negative Binomial R.V. given the SUM of its values

Suppose $\{z_{ij}\}$ are independent Negative Binomial random variables with means $\{\mu_{ij}\}$, with $i=1\dots I$ and $j=1\dots J$. How do you find the (expectation of) conditional probability ...
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1answer
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Conditional variance and expectancy of two independent poisson variables

Le $\ X \sim Pois(5) , Y \sim Pois(10) $ both independent. Suppose I draw and rectangle with width $\ X $ and length $\ Y $. Suppose the circumference of the rectangle is $\ 28 $ what is $\ Var(X) $ ? ...
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First event distribution of a non-homogenous Poisson Process when the event is not bound to occur

Let $N(t)$ a non-homogeneous Poisson Process with rate $r(t)$. If the rate sum up to infinity, i.e. $\int_0^\infty r(u)du=\infty$, the first event distribution, $f_{T_1}$ can be expressed as: \begin{...
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1answer
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Characteristic function of independent Poisson random variables

Let $X_{i}$ be independent Poisson distributed random variables with parameter $\lambda_{i} > 0$ for $i = 1,\ldots,n$. Now the joint distribution is given by \begin{equation*} \mathbb{P}\left(X_{...
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2answers
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Probability that $25$ calls are received in the first $5$ minutes.

Calls are received at a company according to a Poisson process at the rate of 5 calls per minute. Find the probability that $25$ calls are received in the first $5$ minutes and six of those calls ...
8
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1answer
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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 \quad (*) \end{align} where $X$ is a Poisson random varaible with ...
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Poisson Variable $P(X \le 2)$ - is there a better way?

Let $X$ be a poisson variable with $\lambda=2$. Calculate $P(X\le 2)$. Is there a better way for calculating this rather than the following? $$P(X \le 2)=P(X=0)+P(X=1)+P(X=2)$$
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Determine uniform parameters in mixed Poisson model

I have to determine the continious uniform parameters in a certain Poisson distribution. To make it a bit clearer: $X_i $ ~ $Pois ( \Lambda) $ and $\Lambda $ ~ $Unif (a,b)$ where $a = 0$. Also ...
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1answer
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Exponential and Poisson distribution, machine

I have this question where i am unsure how to solve it. X...how often a machine does not work E(X)= 3 per day= 1/8 per hour X-Poisson distributed What is the probability that no machine breaks ...
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1answer
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$P(2X+2Y=28)$ Poisson Variable Distribution

Let $X$ be a random poisson variable with the parameter $5$, let $Y$ be a random poisson variable with the parameter $10$. $X, Y$ are independent of each other. What is the probability that $2(X+...
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20 views

How to solve this Poisson-distribution question?

The weight of certain type of muffin is approximately 50 grams. Let X be the number of muffins that weigh more than 50 grams has the probability of 0.005. A sample of 100 muffins are selected randomly....
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1answer
31 views

How to solve this problem with Poisson distribution

Problem: A store owner observes that there are $3$ (in average) customers visiting the store per hour. He wants to find the probability that there are at least $1$ customer visiting his store in $...
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1answer
27 views

Compute $\ P\{X_1 = 3 | X = 10 \} $, $\ X \sim Pois(10) $

Suppose $\ X$ is the number of people entering a store in an hour and $\ X \sim Pois(10) $. Compute the probability that at most 3 men enter the store if it is known that 10 women already entered. ...
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28 views

Poisson Process For Large n

A supermarket has two entrances, the main entrance and the side entrance. The arrival of customers through the main entrance follows a Poisson process with rate 3 per minute and the arrival of ...
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1answer
34 views

Problem understanding the math in HyperLogLog

I have problem understanding how the math works in the hyperloglog algorithm. More specifically, I have trouble seeing how the author get formula 5 from formula 4, in the HyperLogLog paper, page 132. ...
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Approximate independence for fixpoints of random permutations

Let $F_n$ be the random variable that is the number of fixed points of a random permutation on $n$ elements. As $n \to \infty$, the distribution of $F_n$ approaches a Poisson distribution with mean 1....
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2answers
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Which distribution do I need to use?

In a shop, the average customers per 5 minutes is 3. What is the probability that the shopkeeper has to wait at least 6 minutes before the second customer walks in. I don't know which distribution I ...
0
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1answer
32 views

probability - vehicle arriving and gap in between them

Vehicle arriving at an intersection from one of the approach roads follow the Poisson distribution. The mean rate of arrival is 900 vehicles per hour. If a gap is defined as the time difference ...
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1answer
55 views

Strong law of large numbers for Poisson rvs with different parameter

Let $X_n$ be independent Poisson random variables with $E[X_i] = \mu_i$, and let $Y_n = X_1+...+X_n$. I want to show that if $\sum_n \mu_n = \infty $ then $Y_n/E[Y_n] \rightarrow 1$ almost surly. What ...
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A question about Poisson Process: operating events from different sample spaces?

The following proof, that how to derive Poisson Distribution from a Poisson Process, is from my textbook, Elementry Probability Theory(Fourth Edition), written by Kai Lai Chung, Farid AitSahlia. ...
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1answer
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“Following” $\operatorname{exp}(\lambda)$ random variables “sum” to $\operatorname{Poi}(\lambda t)$ random variable

Lifetime of a bulb is distributed $\operatorname{exp}(\lambda)$. When one light bulb is burned we replace it immidietely. Let $N_t$ be the number of bulbs we've used by time $t$. Prove that $N_t \sim \...
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Choosing a parameter for a Poisson distribution

This question is kind of weird, and it raised from an even weirder question (in the bins and balls model), but I've tried to simplify it as much as I can let $n,k\in \mathbb N$ be 2 numbers such that ...
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1answer
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Poisson adaptation of IRT model: how to interpret parameters?

for my work I made an assessment in which users have to stack blocks in a certain configuration in as few steps as possible. If the user does this in the minimally required number of steps, they get ...
0
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1answer
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Poisson Event After Time Interval

So I have this Poisson Problem that I'm struggling with, and the basis is that you have a server that fails once every four hours (so the average is 1/4 of a crash per hour). The question that I'm ...
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2answers
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How to calculate $\mathbb{P}(|U_{1} - U_{2}| < \frac{1}{12})$, where $U_{i} \sim \mathcal{U}(0, \frac{1}{2})$ and $U_{i}$ are independent.

We have a Poisson process of intensity $\lambda = 4$. We have the following event: $A$: "Two marks appear with a separation of $\frac{1}{12}$ or less". We need to calculate the probability of ...
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1answer
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Compute the Poisson distribution

I have the following problem: " In a city two serious accidents happen per week on average. In particular, we assume that the number of serious accidents is Poisson distributed. Calculate the ...
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How can you prove time between random Poisson points is exponential?

How can you prove time between Random Poisson Points is exponential? I know you can derive the Poisson Distribution from exponential but i'm not sure if this is enough to prove about the time ...
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2answers
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Computing $\ E[Y^2] $ when $Y$ is a piecewise function of $X \sim Pois(2) $

$\ X \sim Pois(2) $ and let$\ Y $ be a random variable : $\ Y = \begin{cases} 2X , X \le 3 \\ X , X \ge 4 \end{cases} $ compute $\ E[Y^2] $ ? I have calculated that $\ E[Y] = E[X] + 10e^{-2} $ and ...
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Properties of Poisson Processes Further Investigated

I am reading Introduction to Probability Models by Sheldon M. Ross, and I am having a difficult time comprehending this example. The text explains this section 'Further Properties of Poisson Processes'...
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Distribution of the number of outcomes from a Poisson distribution with two splits

I expect guests for my birthday party. Their number follows a Poisson distribution with some mean $\lambda$. With probability $p$ any arriving guest will be left-handed and with probability $1-p$, ...
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2answers
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$E(X!)$ for Poisson distribution [closed]

For $X\sim\text{Pois}(λ)$, find $E(X!)$ (the average factorial of $X$), if it is finite. Solution: By LOTUS, $$E(X!) =e^{−λ}\sum_{k=0}^{\infty} k!\frac{\lambda^k}{k!} = \frac{e^{−\lambda}}{1−\...
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Expectation and variance of number of movie tickets

Let $N\sim\mathrm{Pois}(\lambda_1)$ be the number of movies that will be released next year. Suppose that for each movie the number of tickets sold is $T\sim\mathrm{Pois}(\lambda_2)$, independently. ...
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1answer
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Problem of statistical inference Poisson

I am having problems solving this problem of statistical inference and I do not know if it is well done or not, so I would like someone to review it. I just started with inference, so I have my ...
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1answer
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Markov’s inequality and Poisson distribution [closed]

Let X be a random variable having the Poisson distribution with parameter 1. What does Markov’s inequality (p.72) imply about the probability P{X ≥ 2}?
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Calculating necessary assumptions on simple Poisson process

I'm reading a textbook on probability theory, and the author has recently introduced the Poisson process. We're counting occurrences in an interval of time; this quantity is a random variable for each ...
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1answer
28 views

Poisson Distribution Variance Problem

Here is the question- The number of computer servers that break down during a month is a Poisson Random Variable with parameter $\lambda = 2$. The cost of repairing one server is 2000 and also there ...
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2answers
48 views

Distribution of no. of siblings of a random child if the no. of children of a family is Poisson distributed

Consider a large population of families, and suppose that the number of children in the different families are independent Poisson random variables with mean $\lambda$. Show that the number of ...
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Log-likelihood of zero-truncated Poisson

Question: Edwards and Eberhardt (1967) conducted a live-trapping study on a confined population of known size. In their study, wild cottontail rabbits were penned in a 4-acre rabbit-proof enclosure. ...
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Expectation of sum of independent Poisson distributions

I have three independent Poisson distributions: $X_1 \sim \mathcal{P}(15)$, $X_2 \sim \mathcal{P}(21)$ & $X_3 \sim \mathcal{P}(10)$. I wish to find the Expectation and Variance of $X_1 + X_2 + ...
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Maximum likelihood estimation in a Poisson convolution

Suppose that $X$ and $Y$ are i.i.d. Poisson random variables, with mean $\nu$. The parameter $\nu$ is unknown and we would like to estimate it. We only are given the single data point $$ X-Y. $$ What ...
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1answer
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Intuition behind Expected Value of the Square of a Random Variable $X$

Let's look at the case of $X \sim Pois(\lambda)$. Since $k \in X(\Omega)$, it is clear that $k^2 \in X(\Omega)^{2}$. Following this logic and from an intuitive view I'd say $\mathbb E[X^{2}]=\sum_{k^...
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$\mathbb{E}(X_{Y+1}X_{2}^{2}X_{2}|x_{1})$ with $X\sim N(0,1)$ and $Y\sim Pois(1)$ both independent

Let $\{X_{i},i\in\mathbb{N}\}$ be a sequence of independent standard normal random variables. Furthermore, $Y$ is a Poisson distributed random variable with parameter $\lambda=1$, i.e., $\mathbb{P}(Y=...
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On a Poisson distribution

Here is the Question I'm stuck on: A petrol station has service areas on both sides of a motorway, one to serve north-bound traffic and the other for south-bound traffic. The number of north— bound ...
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Poisson process detected with prohability p

I have a question about a homework exercise of my statistics class, concerning poisson processes: Consider a poisson process with rate $\lambda$. An event is detected only with the prohability $p$....
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1answer
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Finding the conditional distribution of a poisson process

This question is from a workbook i'm currently working on. If we have a poisson process thats is on a real line and denote it with $S(x_1,x_2)$ as the number of events in the time interval between $...
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1answer
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Number of events occurring in a time period is a Poisson with parameter $\lambda$.

Suppose that the number of events occurring in a time period is a Poisson random variable with parameter $\lambda$. If each event is classified as a type $i$ event X with probability $p_i$, $i = 1,... ...
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Departure rate of Poisson tasks with aggregation

There is a single class of tasks with Poisson arrival rate λ at a processing node which takes a fixed-length time interval of D seconds to process a task. These tasks could be aggregated in a way that ...
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Probability Question Poisson Distribution with Game Theory

The Question: Suppose that you receive emails at an average rate of 6 per hour. At 10:00am, your inbox is empty. At 10:10am, you receive a message on your computer that you have received at least one ...
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System reliability: Can we say anything about the distribution of UP and DOWN times?

On page 620, example 9.32 of Introduction to probability models, the average UP and DOWN times of a system that is composed of some components is given for a series system (all components should ...