For a continuous predictor $x$ and a categorical factor, an additive Poisson regression with log link has $\log {\mu }_j(x)=a_j+bx$. The log-mean curves are parallel and ${\mu }_j(x)/{\mu }_k(x)=e^{a_j-a_k}$ is constant in $x$. Adding a predictor-by-factor interaction gives $\log {\mu }_j(x)=a_j+b_{j}x$, so this ratio becomes $e^{a_j-a_k+(b_j-b_k)x}$. With $J$ factor levels, the common-slope restriction has $J-1$ independent constraints. A likelihood-ratio test compares the two deviances with an asymptotic chi-squared distribution on $J-1$ degrees of freedom. Evidence against common slopes establishes varying effects on the log scale; it does not by itself determine their signs or the ordering of the mean curves.

For independent aggregate Poisson distributions with means $n\lambda$ and $n\lambda e^{\beta }$, conditioning on the total eliminates $\lambda$. Independent Poisson conditioning gives the displayed binomial distribution. Its conditional likelihood is proportional to $e^{\beta Y_1}(1+e^{\beta }{)}^{-m}$, also the profile likelihood after maximizing over the unrestricted positive baseline rate. Unequal exposures replace the binomial success probability by $n_1{e}^{\beta }/(n_0+n_1{e}^{\beta })$.

For independent Poisson distributions with log means $Z\beta +o$, where $o$ is a known offset, differentiation of the log-likelihood gives the score function $Z^T(y-\mu )$. An interior maximum-likelihood estimator therefore satisfies the displayed equations. An intercept and categorical indicator columns make fitted count totals equal observed count totals in each represented factor level. These equations match counts, rather than unweighted averages of rates when exposures differ.

// Target: probability-and-statistics.bigb

In Poisson regression with log means $a+\beta x_i$, the Fisher information matrix is $\left(\begin{array}{cc}{S}_0& S_1\\ S_1& S_2\end{array}\right)$, where $S_j={\sum }_i{x}_i^j{\mu }_i$. Inverting this matrix gives slope variance $S_0/(S_0{S}_2-S_1^2)$. Writing ${\bar{x}}_{\mu }=S_1/S_0$ reduces it to the displayed reciprocal weighted sum of squares. Thus a common intercept consumes information about the overall rate, while slope information comes from the spread of the covariates under the mean-count weights. Constant covariates make the two parameters unidentifiable.

For a full-column-rank design $X$ and positive fitted means ${\mu }_i=\exp (x_i^T\beta )$, the log-likelihood has Hessian $-X^T\mathrm{diag}({\mu }_i)X$, which is negative definite. Thus any finite stationary point is the unique maximum-likelihood estimator. Full rank does not guarantee existence: with an intercept and all counts zero, the supremum is approached as the intercept tends to $-\infty$. If every observed count is positive, a full-rank design does give a finite maximum: each function $y_i{\eta }_i-e^{{\eta }_i}$ tends to $-\infty$ in either tail and is bounded above, and $\Vert X\beta \Vert \to \infty$ as $\Vert \beta \Vert \to \infty$, so the negative log-likelihood is coercive.

For independent counts with means $p_{i1}{\lambda }_i$ and $p_{i2}{\lambda }_{i}r$, let $t_i=y_{i1}+y_{i2}$. Maximizing the Poisson regression likelihood in each baseline rate gives the displayed estimate. The remaining log-rate-ratio score isEquivalently, conditional on $t_i$, the second count has a binomial distribution with success probability $r{p}_{i2}/(p_{i1}+r{p}_{i2})$. Each positive-total term increases strictly with $r$; if both treatment totals are positive, the score has a unique positive root. This reduces a many-parameter fit to a scalar equation.

The geometric series gives $Q_r(t)=2{\sum }_{n\ge 1}{r}^n\sin nt$. Its convolution multiplier is $-i\mathrm{sgn}(n)r^{|n|}$ for nonzero frequencies and zero at frequency zero. Since the series is uniformly absolutely convergent for $r<1$, these identities hold for every Lebesgue integrable function by termwise integration. If all negative Fourier coefficients vanish, $Q_r\ast f=-i(P_r\ast f-\hat{f}(0))$.

The Poisson kernel on the circle is nonnegative, has normalized integral one, and its mass outside any fixed arc about zero tends to zero as $r\uparrow 1$. Split the convolution error into a short arc, controlled by the modulus of continuity, and its complement, controlled by the vanishing kernel mass. This proves uniform convergence for every continuous periodic function. Its Fourier multiplier is $r^{|n|}$, so the convolution is the Abel sum of the Fourier series.

For a meromorphic function near $0<|z|\le 1$, and a target $a$ off its image of the unit circle, let $u_a=-\log k(f,a)$ for the chordal metric, $M_a(s)$ its normalized angular mean, and $I(a)=M_a^{\prime }(1)$. The weighted spherical area $T(r)=(4\pi {)}^{-1}{\int }_{r<|z|<1}\log (|z|/r)(f_{\mathrm{round}}^{\#}{)}^{2}dA$ satisfies the displayed formula, where $N$ counts $a$-points with multiplicity and weight $\log (|z|/r)$ and $\overline{m}=M_a(r)-M_a(1)$. Indeed $\Delta u_a=(f_{\mathrm{round}}^{\#}{)}^2/2-2\pi \sum {\mathrm{deg}}_z(f){\delta }_z$, and Green's second identity with the radial logarithmic weight proves the formula. The angular mean must include $1/(2\pi )$.

If the positive baseline rate has a gamma distribution of shape $a$ and rate $b$, its logarithm has this probability density function with respect to $d\mu$. The extra factor $e^{\mu }$ is the Jacobian determinant of the transformation. After a Gibbs sampler draws the conditional baseline rate, taking its logarithm is an exact update of the intercept. Omitting the Jacobian changes the prior.

Let independent counts satisfy $Y_i\sim \mathrm{Poi}(\theta w_i)$, with positive known exposures $w_i$, $\theta >0$, and $a_n={\sum }_{i=1}^n{w}_i\uparrow a_{\infty }<\infty$. Their likelihood is proportional to ${\theta }^{S_n}{e}^{-a_n\theta }$, where $S_n={\sum }_{i=1}^n{Y}_i$. If $S_n>0$, the positive maximum-likelihood estimator is $S_n/a_n$; when $S_n=0$, no maximum is attained on the open parameter space, while the closed-space maximizer is zero. This extended estimate converges almost surely to $S_{\infty }/a_{\infty }$, where $S_{\infty }\sim \mathrm{Poi}(\theta a_{\infty })$. Finiteness follows from $\mathbb{E}{S}_{\infty }=\theta a_{\infty }$, and the limiting Poisson distribution follows from the finite-sum laws. In particular, the probability of a zero estimate tends to $e^{-\theta a_{\infty }}>0$, proving failure of statistical consistency. The total Fisher information tends to $a_{\infty }/\theta$: infinitely many observations need not provide infinite information.

A rate ratio compares event counts per unit exposure between two conditions. In a Poisson regression with a log link and binary indicator coefficient $\beta$, it equals $e^{\beta }$. Exponentiating an approximate Wald confidence interval for $\beta$ gives an interval for the rate ratio. A rate ratio is distinct from a relative risk or an odds ratio.

The Poisson equation for differential forms on a closed oriented Riemannian manifold is solvable exactly when its source is $L^2$-orthogonal to every harmonic differential form of the same degree. Necessity follows from the formal adjoint identity $⟨\Delta \alpha ,h⟩=⟨\alpha ,\Delta h⟩=0$. For sufficiency, the Green operator of the Hodge Laplacian gives $\alpha =G\beta$, since $\Delta G=I-P_{\mathcal{H}}$.

For $N$ with Poisson distribution of mean $\lambda$, the equality $n\mathbb{P}(N=n)=\lambda \mathbb{P}(N=n-1)$ proves the identity by summation. For an independent random sum of independent claims $S$ with this count, exchangeability of the summands gives $\mathbb{E}[Sh(S)]=\lambda \mathbb{E}[Xh(S+X)]$, where $X$ on the right is independent of $S$ and has the severity law. These identities hold for nonnegative measurable functions, and for integrable signed functions. They obtain moments without assuming that a positive exponential moment exists.

The Stein-Chen method compares an integer-valued count with a Poisson distribution by solving the Poisson Stein equation and estimating its expectation at the count. For sums of indicator random variables, dependency neighborhoods expose local overlap errors. This gives total variation distance bounds without having to compute the entire count distribution.

The information entropy $F(\lambda )$ of a Poisson distribution is strictly increasing for $\lambda >0$. Given ${\lambda }_2>{\lambda }_1$, take independent $U\sim \mathrm{Pois}({\lambda }_1)$ and $V\sim \mathrm{Pois}({\lambda }_2-{\lambda }_1)$. Their sum has Poisson distribution with mean ${\lambda }_2$, so entropy monotonicity under independent addition gives $F({\lambda }_2)\ge F({\lambda }_1)$. All these information entropies are finite, and $\mathrm{Cov}(U+V,V)=\mathrm{Var}(V)>0$ excludes equality.

// Target: probability-and-statistics.bigb

A mixture of Poisson distributions over a nonnegative random intensity. With exposure $n>0$ and prior density $\pi (\theta )$, its probability mass function is $p_n(s)={\int }_0^{\infty }{e}^{-n\theta }(n\theta {)}^s\pi (\theta )d\theta /s!$. A gamma mixing distribution gives a negative binomial distribution. Counts sharing the same latent intensity are conditionally independent, but generally dependent after mixing.

The distribution of $N_{+}-N_{-}$ for independent Poisson random variables with means ${\lambda }_{+},{\lambda }_{-}$ is the Skellam distribution. Its characteristic function is $\exp ({\lambda }_{+}(e^{iu}-1)+{\lambda }_{-}(e^{-iu}-1))$, its mean is ${\lambda }_{+}-{\lambda }_{-}$ and its variance is ${\lambda }_{+}+{\lambda }_{-}$. For equal means $\lambda >0$,This is the fixed-time law of a symmetric Poisson difference process.

// Target: markov-process.bigb

For one acoustic half-space and vacuum on the other side, a harmonic point force gives displacement amplitude $F\gamma /D$ and fluid acoustic velocity potential amplitude $i\omega F{e}^{-\gamma y}/D$. For time dependence $e^{-i\omega t}$, the outgoing acoustic square-root branch has $\gamma >0$ for $|k|>k_0$ and $\gamma =-i\sqrt{k_0^2-k^2}$ for $|k|<k_0$. The causal limit $\omega \mapsto \omega +i0$ places the positive guided-mode pole above the inversion contour and the negative pole below. The saddle point generates cylindrical radiation, whereas guided residues decay normally to the membrane.

For the Boussinesq top-hat plume in a stratified ambient with $N^2=C{z}^p$, the exact pure-power solution has the displayed radius and velocity, and $g^{\prime }=-2C{z}^{p+1}/(3p+8)$. With positive entrainment, radius, upward speed and reduced gravity, stable $C>0$ requires $-4<p<-8/3$, whereas unstable $C<0$ requires $p>-8/3$. The stable branch has divergent buoyancy flux at zero height, so it is not a finite-flux point-source solution. The unstable branch has zero limiting source buoyancy flux and extracts buoyancy from its maintained unstable ambient. The cases $p=-6,-4,-8/3$ are singular at nonzero $C$; $C=0$ instead admits the ordinary Boussinesq point-source plume.