Articles of poisson distribution

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 […]

Show CLT for Poisson random variables, using no generating function

Question is as following: $X\sim Po(\lambda)$ $$\frac{X-\lambda}{\sqrt{\lambda}} \,{\buildrel d \over \rightarrow}\, N(0,1)$$ as $\lambda \rightarrow \infty$. Obs. One is asked not to show the convergence with generating functions. I begin by setting $Y=\frac{X-\lambda}{\sqrt{\lambda}}$ and thus: $P(Y\leq y)=P(\frac{X-\lambda}{\sqrt{\lambda}} \leq y)$. After some arrangements: $P(X \leq y\sqrt{\lambda}+\lambda)$. The plan after this is to use the cdf of […]