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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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How to transform a $U(0,1)$ variable to produce a Poisson variable?

Suppose $ X $ is a uniformly distribution over $(0,1)$. How to find transformations $Y=g(X)$ to produce random variables with the Poisson distribution?
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Transforming sum of n exponential distribution to a Poisson distribution

Let $X_1,...,X_n$ be i.i.d exponential random variable with mean $\lambda$ $S=X_1+...+X_n$ So by finding the mgf of S, we get that $S \sim \operatorname{Gamma}(n,\lambda)$ The problem I am stuck ...
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let $X$,$Y$ be independent Poisson distributed random variables with parameter $\alpha$ and $\beta$ respectively. $E(XY)$?

$$E(XY) = \sum_{x,y} xy f(x,y),$$ but I don´t have the $f(x,y)$. $X+Y$ would be Poisson distributed with parameter $\alpha + \beta$, but what about $XY$? Not sure what else to do. Thanks in advance.
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Is the zero truncated Poisson Distribution part of the Exponential Family?

This is the density of a truncated Poisson: $$P(X = x \mid X > 0) = \frac{\lambda ^ x e^{- \lambda} }{x ! \left ( 1 - e^{- \lambda} \right )}$$ To show that it's member of the Exponential ...
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Difference between Poisson processes and Poisson distribution

We suppose that a factory has on average 3 call per minutes. What is the probability to have 3 call in 4 minutes? I'm always confuse. Should I use a Poisson random variable or a stochastic process? i....
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Proof for Void probability of a Binomial Point Process

We need to prove: $$P[N(B)=0|N(A)=n]=\left(1-\frac{|B|}{|A|}\right)^n$$ The attempt: Let $\bar{B}=A\text\B$ \begin{align}P[N(B)=0|N(A)=n]&=\frac{P[N(B)=0\bigcap N(A)=n]}{P[N(A)=n]} \\\\ & =...
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Why $\mathbb P\{S_n=k\}\approx \frac{\lambda ^k}{k!}e^{-\lambda }$ if $S_n\sim Binom(n,p)$? [on hold]

Let $S_n\sim \textrm{Binom}(n,p)$. A theorem (without proof), says that if $n$ is big enough $$\mathbb P\{S_n=k\}\approx \frac{\lambda ^k}{k}e^{-\lambda },$$ where $\lambda =np$. Honestly, I don't get ...
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Compute the profile log likelihood for $\alpha$ giving your answer in terms of $\hat{\alpha}_{\beta}$.

We have $X_{1}, X_{2},...,X_{n}$ IID random variables from a Poisson distribution with mean $\mu_{i}=\exp{(\alpha + \beta z_{i})}$. i) For fixed $\beta$, find $\hat{\alpha}_{\beta}$, the maximum ...
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Bulbs with amnesia

Here is a question for which I am not able to figure out the approach to solving it. Problem statement: Suppose that $n$ light bulbs in a room are switched on at the same instant. The life time of ...
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Conditions for the Poisson distribution

Two problems that are related that I'm stuck on, any help greatly appreciated: Q1: During the football season, an amateur football club holds training sessions for its first team squad on Tuesdays ...
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Having trouble interpreting this experiment on the Poisson distribution

von Bortkiewicz considered the frequency of deaths from kicks in the Prussian army corps. From the study of 14 corps over a 20-year period, he obtained the data shown in the table below. Fit a ...
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Poissonian nature of photon count

I am trying to use poissonian distribution to validate photon emission of x-ray source. Photons counts are recorded at 100ms intervals using a photon counting detector. If the photon distribution is ...
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Does the sum of the Poisson approximations to a Bernoulli trial never equal one?

The sum of the Poisson distribution is equal to exactly 1 only when the sum of $P(X = j)$ is taken for all values of j (0 to infinity): $$\sum_{j=0}^{\infty} e^{-\lambda}\frac{\lambda^j}{j!} = 1$$ ...
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A hard Poisson distribution problem that I can't get my head around

Question : " Customers arrive at a shop such that the number of arrivals in any interval of duration d hours follows a Poisson distribution with mean 8d. The third customer on a particular day arrives ...
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Poisson distribution/ finding special k

Let $X$ be a discrete random variable with its probability mass function: $$p_X(k)= P(X=k) \ for\ k\in \mathbb{N}_0 $$. I want to find $k_0$,such that $p_X(k_0) \geq p_X(k) \forall k \in \mathbb{N}...
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Devising a hypothesis test for machine failure rate

I'm trying to devise a hypothesis test for failure rate data of machines. The gist is that there are some machines in a factory that run all the time. They fail from time to time and are promptly ...
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Deriving the Poisson from the Binomial

In my notes I have the following explanation: The probability function of the Poisson random variable is $P_X(k)={\alpha}^k \frac{e^{-\alpha}}{k!}$ A Poisson random variable with parameter $\alpha$. ...
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Almost sure convergence of $\text{Poisson}(\frac 1n)$ to $0$

Let $X_n$ a sequence of random variables such that $X_n\sim \text{Poisson}(\frac 1n)$. Study the almost-sure convergence of $X_n$. Since $X_n$ is integer-valued and $P(X_n=0) = \exp(-\frac 1n)$ it is ...
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Where does poisson distribution functional form come from?

I know that the punctual probability function of a random variable $X$ with a Poisson distribution is: $$P(X=k)= e^{-\lambda }\frac{\lambda ^{k}}{k!}.$$ Also, I've learned that the formula can be ...
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Approximate distribution of sum of squared standardized Poisson variables

Suppose that $X_1, ..., X_n$ are independent and identically distributed Poisson($\lambda$) random variables. What is a good approximating distribution for $\sum_{i = 1}^{200} \frac{(X_i - \lambda)^2}...
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Censored data maximum likelihood using matlab

I am trying to minimise a likelihood function and estimate the parameter value of $λ$ by fitting to the following data. $t$ is the time and $N(t)$ is the population measured at those specific time ...
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Dart board probability using line method with Poisson application

You randomly throw darts at a dartboard, one dart every second. Suppose that every dart independently hits the dartboard at distance X from the center, where X is a Unif[0,30] random variable. Your ...
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Distribution of arrival times of Poisson point processes

Let $(M_{t})_{t\geq 0}$ and $(N_t)_{t\geq0}$ be two independent Poisson point processes with rate $\lambda$ and $\mu$ respectively. Let $\tau$ be the first arrival time for the process $N_{t}$. Find: ...
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Moment generating function of two Poisson distributions

The time between accidents on the Riverfront Bridge follows a Poisson process with a mean time of 40 days between accidents. The time between accidents on the Overview Bridge follows a Poisson ...
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Is a Poisson r.v.'s parameter a rate $\mu$ or a count $\mu t$?

Let's say I want to model the arrivals of some quantity of interest, say customers coming to a store. I know that on average, $\mu$ customers arrive in on hour. My understanding is that if $N$ is the ...
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Poisson Random Variable Question

A radioactive source emits certain particles with a Poisson distribution. The probability of no particle emissions during an hour of observation is $0.4$. What is the probability that the first ...
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A relationship between Poisson distribution and gamma distribution

We define $N(t)$ to be number of events in the interval $[0,t]$. We assume that $N(t) \sim P(\lambda t)$ for $\lambda > 0$. Let $X$ be the waiting time until the $n$-th event, we need to prove that ...
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Poisson Process example from Durret's Probability textbook

I'm struggling to understand some examples related to the following theorem in Durret's Probability: Theory and Examples. The theore is: For each $n$, let $X_{n,m}, 1 \leq m \leq n$ be independent ...
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Asymptotic distribution of sample mean of the sum of two poisson distribution

I'm trying to calculate the asymptotic distribution of the sample mean of the sum of two Poisson distributions. Sample 1 is of size N1, and is from a Poisson distribution with expectation $\mu_1$. ...
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differentiating(?) Poisson distribution

I've been facing this - i don't even know how to call it - problem for a few hours now and I have know idea how to "do" this. I mean... I feel like this has something to do with binomality of Poisson ...
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Show that the only unbiased estimator for the zero-truncated Poisson distribution is absurd

Consider the zero-truncated Poisson distribution on the striclty positive integers, i.e. \begin{align} \mathbb{P}_{\theta}(X=k) = \frac{\theta^k}{k!(e^{\theta}-1)}\, \, \, , \, \, k=1, 2, ... \end{...
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Are these Poisson-related problems and are the solutions correct?

In a city there are three kinds of subway lines: the red, green and orange lines. Subways on each line arrive at a station according to three independent Poisson processes. On average, there is one ...
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multiple distributions

Anybody can solve this question that will help me a lot. Losses due to earthquakes in a specific region are distributed uniformly in ($1MM, $5MM) and also number of earthquakes is distributed ...
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Probability-Poisson [closed]

A variable $M$ has a distribution $P(M=k)=\frac{e^{-\beta}\beta^k}{k!}$ for $k=0,1,2,\ldots$. Variable $N$ has a distribution Poisson, $P(N=k)=\frac{e^{-\alpha}\alpha^k}{k!}$. Let $S_1=X_1+ \cdots +...
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Probability and Statistical Modelling Proof (Negative Binomial into Poisson)

The Negative Binomial RV $X$ models the number of trials until the $r$-th success in a sequence of independent Bernoulli Trials with probability of success $p$ in each trial. So, if $q = 1 - p$, $$P(...
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Probability 33 people in a group of 100,000 have the same birthday? Assuming years have 365,000 days.

Probability 33 people in a group of 100,000 have the same birthday? Assuming years have 365,000 days. Would this Poisson Formula work? Updated! Note to OP: Probability of exactly $33$ people have ...
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Finding an unbiased estimator of $(1+\lambda)e^{-\lambda}$ for Poisson distribution

If $X_1,X_2,\ldots,X_n\sim \mathrm{Pois}(\lambda)$, find an unbiased estimator of $(1+\lambda)e^{-\lambda}$. I am actually supposed to find the UMVUE of $(1+\lambda)e^{-\lambda}$. but I first have ...
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Problem about non-homogeneous poisson process

$\textbf{Problem}$ Suppose customers arrive at a system according to the poisson process with rate $\lambda$. Every customer stays in the system for an exp($\mu$) amount of time and then leaves. ...
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Modelling Sales in a software company: Poisson or simple linear algebra?

Let's assume I have 1 year of weekly sale data for software A and 1 year of weekly sale data for software B. Software B is related to software A, because it's a maintenance/security upgrade, so the ...
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What statistical test solution should I use?

I tested an older software and I found one fault per 10 hours run/10 times run. New version of the software has created. How can I prove the software is operating correctly. How many test should I run?...
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Poisson from Exponential

The probability distribution of time between events following a Poisson point process with parameter $\lambda$ is an Exponential Distribution with parameter $\lambda$. The proof from Poisson to ...
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Assume X and Y are independent Poisson random variables with mean 1 and 3 respectively. Find Find P(X≤Y|X+Y =1)

I am having trouble finding P(X≤Y|X+Y =1). I know a conditional probability is going to be (P(X≤Y)AND P(X+Y =1)/ P(X+Y =1). I am unsure how to find these separate variables in the multivariable ...
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Probability question on Arrivals of buses at a bus stop follow a Poisson process of arrival rate [closed]

Arrivals of buses at a bus stop follow a Poisson process of arrival rate $\lambda$ > 0. Each bus has a probability p of being full when arriving at the stop, independent of other buses. A traveller at ...
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Using $X$ in Poisson distributions

Suppose the number of, lets say flowers, are distributed randomly in a field of $1120$ sq. meters. If there are $96$ flowers in this field, then the mean is $96/1120$ for a Poisson distribution. ...
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How will p-values behave when fitting normal/Poisson to binomial?

I know p-values behave uniformly. Now as p(np) is fixed and n goes to infinity, binomial converges to normal(Poisson). Now suppose I take random binomial samplings and fir normal(Poisson) to it, for ...
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Poisson distribution with independent, identically distributed $X_i,$ )

How can I find E($\tilde{X}$) and Var($\tilde{X}$) of a Poisson distribution with independent, identically distributed $X_i,$ )
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Poisson Limit Theorem

Using Poisson Limit Theorem derive an asymptotic probability distribution of n wins (getting 4 correct numbers out of 49) in one lottery (of course many people play the lottery). I only know how to ...
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Distribution of sum of independent Poisson random variables

If $X$, $Y$, and $Z$ are all independent Poisson random variables, each with parameter $\lambda$, can $\mathbb P(X+Y+Z=k)$ be simplified to $\mathbb P(3X=k)$, since $X$, $Y$, and $Z$ are the same? If ...
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Find n from summation

I ran into this expression while reading through a chapter a book: $$\sum_{i=0}^{n}\frac{e^{-1500}1500^i}{i!} \ge 0.95$$ And they solve for $n$ and got $n = 1564$ from the expression, but there's no ...
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Poisson distributed radiation with a faulty counter

There is this problem that I think I have solved. I need feedback if I have solved it correctly. I also have some questions regarding the intuition on the solution I have obtained. Problem Statement: ...