=+24. Consider a homogeneous Poisson process with intensity on the set {(x, y, t) R3

=+24. Consider a homogeneous Poisson process with intensity λ on the set

{(x, y, t) ∈ R3 : t ≥ 0}. The coordinate t is considered a time coordinate and the coordinates x and y spatial coordinates. If (x, y, t) is a random point, then a disc centered at (x, y) starts growing at position

(x, y) at time t with radial speed v. Thus, at time t + u, the disc has radius uv. As time goes on more and more such discs appear. These discs may overlap. This process is called the Johnson-Mehl model.

One question of obvious interest is the fraction of the (x, y) plane that is covered by at least one disc at time t. Show that this fraction equals the Poisson tail probability 1 − e−λv2πt3/3. (Hint: It suffices to consider the origin 0 in R2. The region in R3 that gives rise to circles overlapping 0 at time t is a cone.)