= Solution
Put $L_t=\int_0^t\Lambda(u)\,du$, finite almost surely. Conditional on the intensity process, $N(t)$ is Poisson with mean and variance $L_t$. The <law of total expectation> and <law of total variance> give
$$
\mathbb EN(t)=\mathbb EL_t,\qquad\operatorname{Var}N(t)=\mathbb E[\operatorname{Var}(N(t)\mid\Lambda)]+\operatorname{Var}[\mathbb E(N(t)\mid\Lambda)]=\mathbb EL_t+\operatorname{Var}L_t.
$$
Consequently for a <Cox process>,
$$
\boxed{\operatorname{Var}N(t)\geq\mathbb EN(t).}
$$
These identities apply when the moments are finite; if the mean is finite but the integrated intensity has infinite variance, the count variance is infinite and the inequality still holds.
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