Let $N$ be the PPP on $\mathbb{R}^2$ with mean intensity $$ \mu(dt dx) = e^{-x} dx dt . $$ Sample from $N$ on the box $[-5, 5] \times [-5, 5]$, and plot the result.
A frog is sitting in a rainstorm, and drops are falling on its back at random, independent, Exponential(1)-distributed intervals (i.e., as a PPP(1) in time). Every time a drop lands, it either hops left (probability 1/2) or right (probability 1/2). Let $X(t)$ denote the displacement of the frog after time $t$ (in number of hops relative to its starting location, e.g., $X(t) = -3$ is three hops to the left).
a. Use Poisson labeling to find the mean and variance of $X(t)$ (Hint: let $X(t) = R(t) - L(t)$, where $L$ and $R$ are the number of left and right steps, respectively.)
b. Find the cumulant generating function of $X(t)$, defined as $K_t(u) = \log \mathbb{E}[e^{uX(t)}]$.
c. Use this to derive an expression for the $n$th cumulant of $X(t)$, for each $n \ge 1$. Note that the first cumulant is the mean and the second is the variance.
d. Use simulation to check your result.
We plan to drop a great deal of confetti fom a great height onto a large plaza. Each piece of confetti is a thin paper circle of radius 1cm; suppose that the confetti will fall more or less uniformly across the plaza and that confetti may cover each other.
a. Show that if we drop an average of $\lambda$ pieces of confetti per square meter that the probability a given point on the plaza is uncovered by confetti is $\exp(-\lambda \pi \times 10^{-4})$.
b. What density of confetti (in pieces per square meter) do we need to drop so that around 99% of the plaza will be covered by at least one piece of confetti?
c. Check your solutions by simulation. To do this, simulate and plot a picture of the confetti process on the unit square (to avoid edge effects, simulate uniform confetti on a larger square); then estimate the proprotion covered by asking how often some set of points is covered across many simulations.