What does $X\sim Pois(\lambda S)$ mean?

Question

I'm wondering what exactly does $X\sim Pois(\lambda S)$ mean when $S$ is a random variable as well?

I guess $\mathbb{P}(X=k)=\frac{(\lambda S)^k}{k!}e^{-\lambda S}$ but still I do not know what that truely means. What if, for example, $S=1\pm\epsilon $ each with chance $\frac{1}{2}$?

Answer 1

The value of $S$ alters the distribution of $X$. If we are handed a particular value for $S$ (for instance $S=s_i$) then the resulting Poisson distribution for $X$ will look like the following:

$$\mathbb{P}(X=k|S=s_i)=\frac{(\lambda s_i)^k}{k!}e^{-\lambda s_i}.$$

If $S$ has a countable number of outcomes then we can use the law of total probability to obtain $\mathbb{P}(X=k)$. This law is given by

$$\mathbb{P}(X=k)=\sum_{i=0}^n\mathbb{P}(X=k|S=s_i)\cdot \mathbb{P}(S=s_i)$$

which in our case yields

$$\mathbb{P}(X=k)=\sum_{i=0}^n\frac{(\lambda s_i)^k}{k!}e^{-\lambda s_i}\cdot \mathbb{P}(S=s_i).$$