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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### is the sum of two poisson distribution independent from the others variables [closed]

so i have this question to answer ...
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### 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 \$\...