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When ${\alpha }_i=\alpha$, the law of total variance givesThis is the quasi-Poisson mean-variance relation with constant dispersion, so the most appropriate printed p-value is $0.0982$ from model2.

For independent claim sizes with common expected value $m$ and variance $v$, independent of the nonnegative integer count $N$, the aggregate satisfies $\mathbb{E}S=m\mathbb{E}N$ and $\mathrm{Var}(S)=v\mathbb{E}N+m^2\mathrm{Var}(N)$. Its moment-generating function is $G_N(M_X(t))$ wherever finite. These identities follow from the law of total expectation and law of total variance.

Replace an estimator by its conditional expectation given selected information. The mean is unchanged and the variance cannot increase, by the law of total variance. In simulation this produces conditional Monte Carlo estimators. Sufficiency is needed for the classical inferential strengthening, but not for the elementary variance reduction identity.

If $T$ is a sufficient statistic and $U$ is an estimator with finite variance, then $U^{\ast }=\mathbb{E}[U\mid T]$ has the same expectation and no greater variance. Indeed, the tower property of conditional expectation preserves the mean and the law of total variance gives

Let $B=0$ with probability $\pi$, and otherwise let $B$ have a Gamma distribution with mean one and variance $\tau$. Given $B$, counts $Y_j$ are conditionally independent Poisson random variables with means $B{\mu }_j$. For $q=1-\pi$, the law of total variance and law of total covariance giveEach component has a zero-inflated negative binomial distribution, but the components are not independent after marginalizing $B$. The zero component is shared by the whole profile. At $\tau =0$, interpret the positive Gamma component as a point mass at one.

If each probability measure ${\mu }_i$ has Poincaré constant $C_i$, then the product measure ${⨂}_i{\mu }_i$ has Poincaré constant at most ${\max }_i{C}_i$. This follows by iterating the law of total variance and applying Jensen inequality to derivatives of conditional expectations.

With structural-zero probability $\pi$, size $r>0$ and count mean $\lambda$, the model assigns $\pi +(1-\pi )(r/(r+\lambda ){)}^r$ to zero and $(1-\pi )f_{\mathrm{NB}}(y;r,\lambda )$ to positive counts. Its expectation is $(1-\pi )\lambda$ and its variance is $(1-\pi )(\lambda +{\lambda }^2/r)+\pi (1-\pi ){\lambda }^2$, by the law of total variance. A logarithmic link function can relate the count-component mean to predictors.