Distance estimation with Poisson processes

Question

Assume that Dunkin Donuts stores are distributed about the Boston area as a spatial Poisson process with rate 100 per square kilometer.

(1) What is the expected (Cartesian) distance between any store and the nearest other store?
(2) What is the expected distance from your current location to the nearest store? (3) What is the probability of having at least one store within a radius of 100 meters from your current location?

Assume that Honey Dew Donut stores are also distributed about the Boston area as a spatial Poisson process with rate 21 per square kilometer.

(4) What is the expected distance from any point to the closest donut store (assuming these are the only two brands)? (5) What is the probability that the closest store is a Dunkin Donuts?

I've been looking left and right but don't know how to answer those questions, i know some of them are about my current location, how would I have to proceed to answer to these specific questions?

Answer 1

Let $R$ be the distance from where you are to the nearest donut shop. Then $R$ is a positive-valued continuous random variable. Let $A$ be the event that that nearest one is Honey Dew rather than Dunkin. Then $$ \Pr(A \mid R=r), $$ the conditional probability given that $R$ has a specified value, is just the probability of $A$ given that the number of donut shops in that closed disk is $1.$ Call the two Poisson-distributed random variables $D$ and $H.$ Then \begin{align} \Pr(A\mid R=r) & = \Pr(H=1\mid D+H=1) \\[8pt] & = \frac{\Pr(H=1)\Pr(D=0)}{\Pr(D+H=1)} \\[8pt] & = \frac{(21^1 e^{-21}/1!)(100^0 e^{-100}/0!)}{121^1 e^{-121}/1!} \\[8pt] & = \frac{21}{121}. \end{align} And this does not depend on $R.$ Therefore $A$ and $R$ are independent of each other and this is equal to $\Pr(A).$

Answer 2

Here's an idea: let's say $D$ is the distance from you to the nearest Dunkin Donuts. Then $D > r$ for some $r$ means there is no Dunkin Donuts within a disk of radius $r$ centered at you, so $$P(D > r) = \exp(- 100 \pi r^2)$$ Now use the theorem $$E(D) = \int_0^{\infty} P(D > r) \; dr$$