What an exponential distribution for a spatial poisson process answers I use the Poisson distribution in virology where we try to answer: "What is the probability that X viruses enter a cell given a E(x)=MOI (=virus/cell)".
The analogy is that we throw a certain amount of balls on buckets and we ask how many balls enter in each bucket. If I understand everything correctly this is a case of a spatial Poisson process.
All the explanations about the Poisson & exponential distributions use temporal Poisson process in which:

*

*The Poisson distribution (Pois.dist) is discrete and answers "What is the probability of getting X events in an interval given E(events)" 

*The exponential distribution (Exp.dist) is continuous and answers "What is the probability that X amount of time elapses between m events given E(events/time)
My Question is:

*

*What will be the question that the Exp.dist will answer in the virology case?
Asking a question about the number of cells won't yield a continuous distribution like in the temporal case.

*What is the parameter for the spatial exp.dist?
In the temporal case the parameter for the Pois.dist is a counting and for the Exp.dist is a ratio. While in the spatial case we give a ratio (virus/cell) to the Pois.dist.

*If this isn't a spatial Poisson process, what kind of process is it?

Thanks!
 A: 
The analogy is that we throw a certain amount of balls on buckets and we ask how many balls enter in each bucket. If I understand everything correctly this is a case of a spatial Poisson process.

Ideally, in your infection experiment you have thoroughly mixed the virus into the tissue-culture medium so that all cells have the same probability of being infected. Unless the cells are evenly spaced on the tissue-culture dish, however, emphasizing the spatial aspect by calling it a spatial Poisson process doesn't capture what's happening. Presumably each cell has the same probability of being infected so that its position in space doesn't matter.
The Poisson parameter here is thus the average number of viral particles per cell, perhaps adjusted for a measure of infectivity. Your model is of infection numbers per cell, not of a distribution of infections over the surface of the tissue-culture dish.
There is no obviously useful way, however, to represent this infection of cells on a tissue-culture dish as a point process on the real line. Only that one-dimensional, continuous situation has a natural relationship between the frequency of points along the line and an exponential distribution of the distance/time between them. See the Wikipedia description of a Poisson process "Interpreted as a point process on the real line": there is "no natural equivalence when the Poisson process is defined on a space with higher dimensions."
You don't need an equivalence to the one-dimensional situation, however, for the Poisson distribution to be a useful description of the distribution of viral infection numbers per cell. That's what you care about for designing your experiment.
