# Questions tagged [poisson-distribution]

For questions relating to Poisson distributions in probability theory. To be used with [probability] or [probability-distributions] tag.

1,339 questions
1answer
26 views

### 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?
1answer
35 views

### 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 ...
1answer
24 views

### 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.
0answers
27 views

### 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 ...
2answers
28 views

### 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....
0answers
14 views

### 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]} \\\\ & =...
1answer
98 views

### 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 ...
0answers
7 views

### 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 ...
1answer
24 views

### 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 ...
0answers
30 views

### 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 ...
1answer
18 views

### 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 ...
1answer
32 views

### 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 ...
1answer
35 views

### 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): $$\sum_{j=0}^{\infty} e^{-\lambda}\frac{\lambda^j}{j!} = 1$$ ...
1answer
33 views

### 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 ...
0answers
18 views

0answers
18 views

### 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 ...
1answer
14 views

### 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 ...
1answer
26 views

### 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: ...
1answer
58 views

### 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 ...
1answer
22 views

### 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 ...
2answers
39 views

### 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 ...
1answer
40 views

### 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 ...
0answers
19 views

### 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 ...
1answer
47 views

### 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 $\mu_1$. ...
1answer
30 views

### 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 ...
0answers
37 views

### 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{...
1answer
13 views

### 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 ...
0answers
18 views

### 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 ...
1answer
41 views

1answer
22 views

### 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: ...