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Let $(X_n)_{n \geq 1}$ be a sequence of independent and identically distributed random variables such that $X_n \sim Poisson(\lambda)$. Let $T$ be another random variable, independent of $(X_n)_{n \geq 1}$, and such that $T-1 \sim Geometric(p)$ i.e. $\mathbb{P}(T=n) = p(1-p)^{n-1}$ for all $n \in \mathbb{N}_{>0}$. We define:

$Y:= \sum_{k=1}^T X_k$.

I have to compute the characteristic function of Y.

I don't understand what the sum up to $T$ should mean since $T$ is another random variable?

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  • $\begingroup$ If $\mathbb P\{T=n\}=p(1-p)^{n-1}$, then $T\sim Geometric(p)$ not $T-1\sim Geometric(p)$. $\endgroup$
    – Surb
    May 4, 2021 at 13:15

1 Answer 1

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Hint

Since $$\mathbb E[e^{itY}\mid T=n]=\mathbb E[e^{itX_1}]^n=\exp\left\{n\lambda (e^{it}-1)\right\},$$ you have that $$\mathbb E[e^{itY}\mid T]=\exp\{T\lambda (e^{it}-1)\}.$$

Therefore $$\varphi _Y(t)=\mathbb E\left[\exp\left\{T\lambda (e^{it}-1)\right\}\right]=...$$

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