Poisson distribution process Meteorites hit the surface of the moon, treat as an infinite plane. The meteorites that have hit the moon over the last 1 million yrs can be modeled as a Poisson process with constant intensity lambda. Suppose that each meteorite leaves a circular crater of random radius, and the craters' radii are i.i.d. from a distribution having density f. Assume that f(r) = 0 for r suciently large. For a bounded (Borel) set A, let V (A) be the area of A that is not covered by a crater from a meteorite that hit in the last million years. Since this is a Poisson process, how can we get the E(Area of A not covered by a crater from a meteorite that hit in the last million years) and the Var(Area of A not covered by a crater from a meteorite that hit in the last million years) I do note that E(X) = 1/p

I am not happy with your notation, but one approach is to look at a ring of inner radius $r$ and outer radius $r+\delta r$ (and so of area about $2 \pi r \delta r$ in the limit), calculate the probability either that the ring is not hit or that it is hit and all the meteorites hitting the ring (reducing to 1 in the limit) leave a crater of radius less than $r$. This is the probability that the centre of the ring is not covered by a crater caused by a hit on the ring.

You now want to multiply all the different ring probabilities together. One way to do this is by adding up the logarithms of the probabilities and then taking the anti-logarithm, so in the limit integrate the logarithms of the probabilities over $r$ and take the anti-logarithm.

That will give you the probability that a particular point is not covered by a crater, and so the expected fraction of $A$ which is not covered by a crater. The variance is harder.