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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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Poisson distribution to normal distribution?statistics

I need to demonstrate why when (lambda)is big enought poisson distribution becomes (aproximation)to normal distribution. Thanks you
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Poisson distribution test using index of dispersion

I have a data set which Im trying to check if it is poisson distributed. I read some posts here and online but they are a bit "heavy" on the statistics for a newbie in statistics like me, so I would ...
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Understanding examples of Poisson Distribution

Sheldon Ross describes the following as examples of random variables that generally obey the Poisson probability law: $1.$ The number of customers entering a post office on a given day. $2.$ The ...
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Are events correlated in a Poisson distribution?

I would like some input on a discussion I am currently having with a colleague regarding some measurements that were done recently. I have some radiation physics data that is processed as a histogram....
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Compute $\ P(X_i | \sum X_i = 45) $ where $\ X_i \sim \mathrm{Pois}$

Let $\ X_i \sim \mathrm{Pois}(10) $ be independent random variables $\ i = 1, 2,3,4,5 $ . What is the probability of $\ X_1 = 9 $ given $\ \sum X_i = 45 $ now $\ \sum X_i \sim \mathrm{Pois}(50) $ ...
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Predicting Race Results knowing winning probability

I would like to model a scenario where I have 4 runners [A, B, C, D] and there chances of winning a race are deemed to be [0.5, 0.3, 0.15, 0.05]. I would like to simulate this race 100 times. Is it ...
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Poisson Distribution question on the odds of something happening if the time interval changes

Given an event X that has a 68% chance of happening within 3 years, what are the odds of that event happening each year? I tried to solve this using a Poisson Distribution but I am confused how my ...
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Poisson with Exponentially Distributed Parameter

I am doing a review for the test, and I have found myself really struggling with the following question: Prove using generating functions that if $$V \sim \mathrm{Poi}(\theta),~\theta \sim \mathrm{...
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Poisson Process/Inter-Arrival Time Question

This question is, "People arrive at a train station in accordance with a Poisson process with rate lambda. At time 0, the train station is empty. At time 10, the bus departs. Calculate the expected ...
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Sum of the Poisson distribution (my solution vs. textbook)

I feel something is wrong, but can't place it: Assume $X_i$ are i.i.d. Poisson distribution with parameter $\lambda$ and define $$Y = \sum_{i=1}^n X_i $$ $$M_X(t) = \exp((e^t-1)\cdot\...
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Poisson distribution and soccer scoring

A soccer player scores at least one goal in roughly half of her games. How would you estimate the percentage of games where she scores exactly three goals? $\textbf{My Attempt:}$ I try modeling this ...
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First passage time for Compound Poisson distribution

Let $S$ follows a Compound Poisson distribution $(S \sim CP(\lambda,F_x(x))$, i.e. $$S = \sum_{i=0}^{N}X_i,$$ where $N\sim Po(\lambda)$ and $X_i \stackrel{iid}{\sim} Exp(1)$. I know that the first ...
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Normal approximation of MLE of Poisson distribution and confidence interval

Let $(X_1,\ldots,X_n)$ denote a random sample from a Poisson distribution with parameter $\lambda$. Maximum Likelihood Estimate of $\lambda$ is given by $\hat{\lambda} = \bar{X} = \frac{1}{n} \sum\...
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A tight bound on $\sum_{k=1} ^\infty \frac {x^k} {(y+k)!}$ on $x,y$

Consider the sum: $$\sum_{k=1} ^\infty \frac {x^k} {(y+k)!}$$ When $y=0$, we can simple calculate the answer $e^{x}$. Now suppose $y>x$. How can I bound the summation? I can bound the summation to $...
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Upper bounds for two events of random variable following Poisson distribution

So $X$ follows Poisson with $L>0$ and $\operatorname{E}[X]=\operatorname{Var}(X)=L$. We are dealing with two events: $A=\{X \leq L/2\}$ and $B=\{X \geq 2L\}$. I have calculated that $P(A) \leq 4/L$ ...
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The probability distribution for the waiting time between two clients.

Suppose you have an office that sells tickets for a concert, and your opening hours are between $8:00$ and $12:00$, and also between $14:00$ and $17:00$. You notice that the arrival of clients ...
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Reverse Poisson Distribution problem

I was recently solving a quiz on Poisson distribution and I encountered this question A call center receives an average of • 4.5 calls every 5 minutes. Each agent can handle one of these calls ...
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What is the probability that two or more engines on an aircraft will fail during a flying period of ten hours?

Suppose an airline with a fleet of four-engine aircraft observed that on average, an engine with normal preventive maintenance failed two times in 10,000 operating hours. Using the Poisson model, What ...
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Rate parameter in Poisson process - confusion regarding terminology

I am trying to understand a derivation of maximum likelihood estimation of intrinsic dimension from a Poisson point process given here https://wiki.math.uwaterloo.ca/statwiki/index.php?title=...
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Maximum likelihood estimate of the parameter in Poisson distribution

Given that the number of goals scored per match by a football team can be assumed to be a Poisson random variable with mean $\theta$. In eight games, the team scores 3, 6, 2, 5, 4, 1, 4, 5 goals. (a) ...
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probability of penalty (in soccer) being scored or being missed

I have the following assumptions. I am expecting 0.4 penalties in a match on average, and I am assuming that penalties follow a Poisson distribution. The probability of a penalty being converted is 82%...
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If $N$ is a Poisson process, then $\operatorname P\left[\exists t>0:N_t=n\right]=1$

Let $(N_t)_{t\ge0}$ be a Poisson process with parameter $\lambda\in[0,\infty)$ and $n\in\mathbb N$. How can we show that $\operatorname P\left[\exists t>0:N_t=n\right]=1$? Assume for the moment ...
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Find critical value given test statistic Poisson distribution

I know how to perform $z$ test and $t$ test on a Normal distribution, but now I'm given i.i.d $X_1, X_2,...,X_n$ drawn from a Poisson distribution. Using $\sum_{i=1}^{n} X_i$ as the test statistic, ...
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Expected value of a Gamma RV to the power of a Poisson RV

$\mathit{W}$ is a $\bigl(\alpha = 3, \beta = \frac 12 \bigr)$ -Gamma random variable, and $\mathit{N}$ is a $\mu$ = $\frac 13$ -Poisson random variable, independent from $\mathit{W}$. What is $\...
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Why does the Poisson Distribution have an exponent in its formula?

The formula for Poisson Distribution is as follows: $$P(X=x) = \frac{e^{-\lambda} \cdot \lambda^x }{x!}$$ Where, $\lambda =$ average (or, mean) rate of an event to take place. In other words, mean ...
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Find the probability that the company will receive exactly k claims from A policies tomorrow.

Consider a large insurance company with two types of policies: policy A and policy B. Suppose that the number of claims the company sees in a given day has a Poisson distribution with parameter $\...
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P(X= 1 | X+Y=9) with Poisson Distribution (X,Y independent)

When trying to calculate the value of $P(X= 1 | X+Y=9)$, I tried doing: $P(X= 1 | X+Y=9) = \frac{P(X=1\cap X+Y=9)}{P(X+Y = 9)} = \frac{P(X=1)P(Y=8)}{P(X+Y = 9)} $, but I'm constantly getting an ...
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The distribution of Y=(-1)^X and the expected value [closed]

I have the X variable which has Poisson distribution and I have to find the distribution of Y= (-1)^X. How should I do this ? Y Has only two values. How diesel its expected value look like?
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Approach to Poisson Conditional Probability

Suppose I have data that is poisson distributed, with event $B$ occurring $\lambda = 2$ per year. Suppose I know $B$ does not occur in the first $6$ months. What is the probability that $B$ occurs ...
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Expectation of special homography for poisson distribution

Let $ N \sim \mathrm{Poiss}(\lambda)$, then for $a > 1 $ Can expectation $\mathbb{E}\frac{1}{a + N } $ be explicitly calculated? I've noticed that if one denotes the desired expectation as $\...
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Poisson Process: Time until next arrival

Question: Suppose that busses arrive at a bus stop as a Poisson process with rate $\lambda$ starting from time $t=0$ (that is, the interarrival time between busses is exponentially distributed with ...
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Computer screen and Poisson approximation

I am struggling with a elementary probability exercice which I don't see how to "translate" it. I have a computer screen with resolution $768\times1024$ pixels. We suppose that pixels are ...
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Confidence interval for Poisson variables

Let $X_{i},...,X_{n}$ be i.i.d. Poisson random variables with parameter $\lambda>0$ I have: $$\bar{X}={(1/n)\sum_{i=1}^n X_i}.$$ Find two sequences $(a_n)_{n>=1}$ and $(b_n)_{n>=1}$ such ...
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Mean and Variance of total number of fishes.

The number of fishes caught by a skilled fisherman each day has a Poisson distribution with mean $2.5$. After $50$ days, what is the mean and variance for the total number of fishes the fisherman ...
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Compensator of a non-homogenous Poisson process

Given a non-homogenous Poisson process $T_n$ with intensity $\lambda(t)$ and the compensator $\int_0^t\lambda(t)ds$ How can I show that $\int_0^t\lambda(s)ds$ is predictable?
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predictable projection of non-homogeneous poisson process

How to prove that the compensator of a non-homogeneous poisson process is the predictable projection of this non-homogeneous poisson process?
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Construction of probability measure for Hawkes process

How to construct a probability measure for the Hawkes process? Like here
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Distribution of inter-arrival time of non-homogenous Poisson Process

What is the distribution of inter-arrival time of a non-homogenous Poisson Process? In other words, if $T_n$ is a non-homogeneous Poisson process with intensity $\lambda(s)$, and $S_{n+1} = T_{n+1} - ...
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What is the minimum variance band of Poisson Distribution?

I am trying to calculate the minimum variance bound of Poisson Distribution. poisson distribution: P(X=x)=(λ^x)/x! e^-λ, were λ is the mean. I got λ/(sum of x), but I am not sure if this is right. ...
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Conditional law of Poisson process

1)What does the notation for $G_n$ and $H_n$ mean? 2)How would $G_n$ and $H_n$ look like for a Poisson process? 3)How to show that $v$ would be the compensator of a Poisson process? Thanks!
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Interarrival Time Distribution of a Poisson Process

For a Poisson Process with parameter $\lambda$ restricted to the interval $[0, 1]$, what is the probability that at least one of the interarrival times (including the time between $0$ and the first ...
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Maximum Likelihood Estimate with different parameters

Suppose that X and Y are independent Poisson distributed values with means $\theta$ and $2\theta$, respectively. Consider the combined estimator of $\theta$ $$ \tilde{\theta} = k_1 X + k_2 Y $$ where $...
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Marginal distribution of $X$ when $X|m \sim Pois(m)$ and $M \sim \Gamma (2,1) $

I am trying to find the marginal distribution of the joint $X$ and $M$ in order to find the probability $$Pr[X = 0,1,2,3]$$ I am given that $X|m \sim Pois(m)$ and $M \sim \Gamma (2,1) $ so I am ...
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Conditional expectation of $\mathbb{E}(X | X^2 + Y^2)$, with Poisson distribution

I have two independent random variables $X, Y$ with Poisson distribution say that $\mu$ and $\lambda$. I want to calculate the conditional expectation $\mathbb{E}\left(X | X^2 + Y^2\right)$. It is ...
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writing a poisson distribution

The number of calls arriving, X is poisson distributed with a rate of 2 per hour. write the distribution of number of calls arriving in 30mins 30 mins = 0.5 hours. poisson distribution -> $X~P_o (...
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poisson random variable + conditional probability

Let Y be the number of calls to a particular hotline within 10 min. Suppose Y is a Poisson random variable with mean of 3. Find the probability that there are at most 4 calls given that there are ...
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Escaping probability for a poisson random walk

Poisson random walk: Let independent random variables $Z_i \sim Pois(\lambda)$. Consider random walk $ S_n = \sum_{i=1}^{n}X_i, $ where $$ X_i = \begin{cases} Z_i &\text{w.p}\; p\\ -Z_i &\...
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How to use Binomial distribution to figure out probability of X number of success per N times?

Let's say a success happens $120$ times per minute, also $120$ times per $60$ seconds ON AVERAGE. And this means $2$ times per second on average. And we assume we can use Poisson distribution to ...
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Closed-form expression for Poisson-Binomial series

I'm interesting in knowing whether there is a closed-form expression for the following series: $\displaystyle\sum_{n\geq1}\frac{1}{n!}\lambda^n e^{-\lambda} \left[{n \choose k}z^k(1-z)^{n-k} \right] $...
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Expected value of falling factorials from axioms of Poisson process

Falling factorial, $(x)_n$, is the product of biggest $n$ terms in factorial, $(x)_n = x(x-1)(x-2)\cdot \ldots \cdot (x-n+1)$. Or the number of ways to color the set of $n$ objects into different ...