Homework assignment 3

Consider a simple point process \( \Phi \) defined in \( W = W_1 \times W_2 \subset \mathbb{R}^2 \), having the intensity function \( \lambda(u), u \in W \). Assume that \( W_1, W_2 \) are compact sets, e.g. closed intervals. Define the projection process \( \Phi_p \) by forgetting the second coordinate of the points in \( \Phi \), i.e. \( \mathrm{supp} \, \Phi_p = \{ x \in W_1: \exists y \in W_2 \; \mathrm{such \; that} \; (x,y) \in \mathrm{supp} \, \Phi \} \).

Determine the intensity function \( \lambda_p(x) \) of \( \Phi_p \).

Hint: take advantage of the Campbell formula.