Linked Questions

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1answer
379 views

How can I prove that $X+Y$ is a Poisson process with parameter $\lambda_X+\lambda_Y$? [duplicate]

For 2 independent Poisson processes $X,Y$, with parameters $\lambda_X, \lambda_Y$ respectively, how can I prove that $X+Y$ is a Poisson process with parameter $\lambda_X+\lambda_Y$? To do this, I ...
0
votes
1answer
19 views

Proof involving Poisson and Gamma distributions for two random variables [duplicate]

Prove that if $X_i \sim \text{Poi}(λ_i)$, $i = 1, 2$, are independent, the sum $X_1 + X_2$ has the Poisson distribution as well. Prove that if $X_i \sim \text{Gamma}(\alpha_i,\beta)$, $i = 1, 2$, ...
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0answers
15 views

Poisson distribution and properties of probability [duplicate]

I need the step-by-step solution for this question. If X and Y are aleatory independent variables with a Poisson distribution whose parameters are $\lambda_1$ and $\lambda_2$, respectively. Show that ...
5
votes
2answers
821 views

Convergence of sum of random Poisson variables with divergent parameter

I'm studying almost-surely convergence and convergence in probability of $S_{n}=X_{1}+\cdots+X_{n},$ where $X_{n}$ is distributed $\mathrm{ Poisson}\left(1/n\right)$ with $n\in\mathbb{N}$ and the ...
4
votes
2answers
229 views

Convergence of Poisson distribution

Let $X \sim \mathrm{Pois}(\lambda)$ and $x_1, \cdots , x_n$ observations following this distribution. I want to derive the analytical solution of the following series: $$l(\lambda):=\lim_{ n \to \...
0
votes
1answer
319 views

Variance of a multiple of a Poisson distribution

a) Say we have two independent distribution variables $X \sim Pois(\lambda)$ and $Y \sim Pois(\mu)$. We know that the sum of the two will be $$X+Y\sim Pois(\lambda+\mu),$$ and hence the variance will ...
0
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1answer
186 views

A problem on queues [closed]

I am confused about how I can use queue theory on this problem. A government has two officers handling people's requests. Suppose at the time between requests arrivals at the first officer's desk is ...
0
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0answers
245 views

The sum of n independent poisson($\lambda_i$) random variables is poisson($\sum_1^n \lambda_i$) random variable. [duplicate]

How can I show that the sum of n independet poisson random variables is a poisson random variable? I need to conclude that the sum of i.r.v with distribution $Poisson(\lambda_i)$, i=1,...,n, is $...
0
votes
1answer
167 views

Conditional expectation of two Poisson variables

In some thesis, while they verified the proposed method, they just mentioned that $$ E[X|X+Y] = (X+Y)\lambda_X/(\lambda_X+\lambda_Y) $$ where X, Y are independent Poisson variables with means $ \...
2
votes
1answer
136 views

Probability of periodically happening event occurring at a given time based on previous data

Assume that an event $X$ happens periodically over time with a period $P_X$. When it starts it lasts for a time $T_X$. $P_X$ and $T_X$ may vary slightly. There is no correlation between $P_X$ and $T_X$...
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1answer
72 views

Confusion with a multivariate Poisson distribution

I am looking at a multinomial Poisson distribution. Failing to obtain the denominator. We have that $X_1, ..., X_n \sim \mathbb{P}(\lambda)$, where $\theta = \exp^{-\lambda}$ We know that $$\mathbb{...
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1answer
67 views
0
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0answers
41 views

Explicit form in a Poisson distribution

What is $\Pr\left( \sum_{i=1}^n X_i \le c \,\middle|\, \lambda = \lambda_0 \right) $ when $X_i \sim \text{Poisson}(\lambda)$? Is it right to say $X_1+\dots+X_n\sim\text{Poisson}(n\lambda_0)$ then $\...
1
vote
1answer
21 views

Arithmetic detail of Sum of Independent Poissons proof

I am stuck with an arithmetic detail of a proof of Sum of Independent Poissons. \begin{align*} P(X+ Y =k) &= \sum_{i = 0}^k P(X+ Y = k, X = i)\\ &= \sum_{i=0}^k P(Y = k-i , X =i)\\ &...
1
vote
1answer
27 views

Probability mass function expression [closed]

What is the expression for the probability mass function for the difference X-c*Y, where X and Y are independent Poisson variables with respective parameters a and b while c is a constant scale factor?...