Articles of poisson distribution

When to use Binomial Distribution vs. Poisson Distribution?

A bike has probability of breaking down $p$, on any given day. In this case, to determine the number of times that a bike breaks down in a year, I have been told that it would be best modelled with a Poisson distribution, with $\lambda = 365\,p$. I am wondering why it would be incorrect […]

Statistics: Deriving a Joint Probability Function From a Definition of Other PDF's

Here’s a particular question I’m trying to understand from the lecture notes. It says: Assume that $Y$ denotes the number of bacteria per cubic centimeter in a particular liquid and that $Y$ has a Poisson distribution with parameter $x$. Further assume that $x$ varies from location to location and has an exponential distribution with parameter […]

Independence of Poisson random variables coming from Poisson sampling

Context: Let $x \in \mathbb{R}^n$ be the unknown probability vector of a finite discrete distribution $X$. We are able to sample $X$ and we want to learn $x$. Poissonization: Each observation belongs to the $i^\text{th}$ category with probability $x_i$, thus for a sample of size $m \in \mathbb{N}$, the sum of the $i^\text{th}$ category follows […]

How to prove Poisson Distribution is the approximation of Binomial Distribution?

I was reading Introduction to Probability Models 11th Edition and saw this proof of why Poisson Distribution is the approximation of Binomial Distribution when n is large and p is small: I can understand most part of the proof except for this equation: I really don’t remember where it comes from, could anybody explain this […]

Crossing the road

A pedestrian wishes to go across a single-lane road where the cars arrive according to a Poisson process with the rate λ. The time needed for him to cross the road safely is denoted as k. He will have to wait until he sees a gap of at least k between the coming cars. If […]

Variance of the random sum of a Poisson?

We have that $N$ and $X_1, X_2, \dots$ are all independent. We also have $\operatorname{E} [X_j] = \mu$ and $\operatorname{Var}[X_j] = σ^2$. Then, we introduce an integer–valued random variable, $N$, which is the random sum such that: $$Z = \sum_{j=1}^{N+1}X_j.$$ Assuming that $N$ is distributed $\sim\mathrm{Poisson}(\lambda)$, what is the first moment and what is the […]

specifying the joint distribution as a proof technique

The following is a theorem in stochastic process in Pinsky’s An introduction to Stochastic Modeling: The proof starts as the following: Here is my question: Could anybody explain why and how one can freely specify the joint distribution of each $\epsilon(p_k)$ and $X(p_k)$ in order to prove (5.11)? The proof continues as the following. But […]

Sums of independent Poisson random variables

The question is the following : $X_n$ are independent Poisson random variables, with expectations $\lambda_n$, such that $\lambda_n$ sum to infinity. Then, if $S_n = \sum_{i=1}^n X_i$, I have to show $\frac{S_n}{ES_n} \to 1$ almost surely. How I started out on this problem, was to consider using the Borel Cantelli Lemma in a good way. […]

How do you sketch the Poisson distribution function: $L(\mu;n)=\frac{\mu^{n}e^{-\mu}}{n!}$?

When doing a maximum likelihood fit, we often take a ‘Gaussian approximation’. This problem works through the case of a measurement from a Poisson distribution. The Poisson distribution results in a likelihood for the average number $\mu$, given that $n$ events were observed:$L(\mu;n)=\cfrac{\mu^{n}e^{-\mu}}{n!}$ Start of question Sketch $L(\mu;n)$. It may also be useful to sketch […]

Find the almost sure limit of $X_n/n$, where each random variable $X_n$ has a Poisson distribution with parameter $n$

$X_{n}$ independent and $X_n \sim \mathcal{P}(n) $ meaning that $X_{n}$ has Poisson distributions with parameter $n$. What is the $\lim\limits_{n\to \infty} \frac{X_{n}}{n}$ almost surely ? I think we can write $X(n) \sim X(1)+X(1)+\cdots+X(1)$ where the sum is taken on independent identical distribution then use the law of large number. But I am not sure that […]