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Suppose there is random variable $K= X + Z$ $$P(X=x)=\frac{T\lambda^{x/\alpha}}{(k/\alpha)!}e^{-T\lambda}\quad(x=0,\alpha,2\alpha,......)$$ $$P(Z=z)=\frac{\lambda^{z/\beta}}{(z/\beta)!}e^{-\lambda}\quad(z=0,\beta,2\beta,......)$$ $$X=\alpha U,\,Z=\beta V \quad (U\,and\,V\,is\,poisson )$$

In case of $\alpha=3,\,\beta=1$, is support of $Z$ [0,1,2,3,.........] right?

I think it is corrrect and i tried to calculate expectation of K.

But it is not $\alpha T \lambda+\beta \lambda$

Kim
  • 179

1 Answers1

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Suppose $U, V \sim \text{Poisson}(\lambda)$.

Then if $X = \alpha U$, $Z = \beta V$ ($\alpha \neq 0, \beta \neq 0$), $$\mathbb{E}[K] = \mathbb{E}[X+Z] = \mathbb{E}[X]+\mathbb{E}[Z]=\alpha\mathbb{E}[U]+\beta\mathbb{E}[V] = \alpha\lambda + \beta\lambda = \lambda(\alpha+\beta)\text{.}$$

Clarinetist
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