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Let $E_i=\{M(I_i)=0\}$. For any subset $K\subseteq \{1,\dots ,n\}$, simultaneous voidness is voidness of the union. Since the intervals are disjoint, the assumed void probability functional givesThis includes the empty subset. To check all event patterns explicitly, apply the inclusion-exclusion principle to the complements in a chosen pattern:Thus the void events are mutually independent, and their complementary occupied-cell indicators are mutually independent too. This is joint independence, not merely pairwise independence.

Draw one random variable $R$, taking values zero and two with equal probabilities. Conditional on $R$, generate a Poisson point process on the real line with constant rate $R$. This is a Cox process directed by the random measure $Rdx$: the same rate is shared by every window. It is a locally finite simple point process and a stationary point process. The law of total expectation gives $\mu (A)=\mathbb{E}[R]|A|=|A|$; the law of total variance and law of total covariance give, for bounded Borel sets $A,B$ with $A\cap B=\varnothing$,The conditional covariance vanishes by independence, but the shared conditional means fluctuate together. This distinguishes conditional from unconditional independence.^[ref]

For a window of length $L$, the exact probability mass function isIn particular, its void probability functional is $(1+e^{-2L})/2$, compared with $e^{-L}$ for the unit-rate Poisson point process. Whole realizations can be empty or comparatively crowded, while their mean density remains one. This example demonstrates shared-rate statistical dependence, rather than a model of short-range offspring clusters.