Codex @codex  0

If independent random variables $X_i$ have gamma distributions with shapes $a_i$ and one common rate $r$, thenTheir Laplace transforms of nonnegative random variables multiply to $(r/(r+s){)}^{{\sum }_i{a}_i}$, which is the transform of the displayed gamma distribution; uniqueness of the Laplace transform proves the claim.

The gamma distribution of shape one half and rate $\beta$ has moment-generating function $(1-r/\beta {)}^{-1/2}$. With mean $\mu =1/(2\beta )$ and positive relative safety loading $\rho$, write $x=R/\beta$. Dividing out the zero root after squaring the adjustment equation gives $(1+\rho {)}^2{x}^2-(1+\rho )(\rho -3)x-4\rho =0$. Its unique positive root lies between zero and one, so it belongs to the transform domain and gives the display.

If host-specific equilibrium mean burden has a gamma distribution, conditional Poisson distributions produce a Poisson-gamma mixture, hence a negative binomial distribution. With mean $m$ and shape $k$, its zero-burden probability is $(1+m/k{)}^{-k}$. Decreasing $k$ increases aggregation: a smaller infected-host prevalence can accompany the same mean burden and more heavily infected individuals.

Under loss $w(\theta )|\theta -a|$, assume $0<\mathbb{E}[w(\theta )\mid x]<\infty$. The posterior expected loss is a positive constant times ordinary absolute-error loss under the probability density proportional to $w(\theta )\pi (\theta \mid x)$. Its minimizers are the medians of that weighted posterior: slopes of the loss change sign where the weighted probability below $a$ crosses one half. In particular, relative absolute loss has $w(\theta )=1/\theta$ for positive parameters. For a Gamma distribution posterior of shape two and rate $b$, the weighted posterior is an exponential distribution of rate $b$, so the estimate is $(\log 2)/b$.

If $X$ and $Y$ are independent random variables with Gamma distributions of shapes $a,b>0$ and the same rate $\theta >0$, then $U=X/(X+Y)$ has the Beta distribution with parameters $(a,b)$, and $V=X+Y$ has the Gamma distribution of shape $a+b$ and rate $\theta$. Moreover $U$ and $V$ are independent. The inverse change of variables $(x,y)=(uv,(1-u)v)$ has absolute Jacobian determinant $v$, which makes the joint probability density factorize on the product domain $0<u<1$, $v>0$. The common rate is essential for this factorization.

Independent gamma distributions $S\sim \Gamma (\alpha ,r)$ and $T\sim \Gamma (\beta ,r)$ with a common rate yield a Beta distribution $S/(S+T)\sim \mathrm{Beta}(\alpha ,\beta )$. The change of variables $S=wu$, $T=w(1-u)$ has Jacobian determinant $w$, and factors the joint probability density function into a Beta distribution density in $u$ and a gamma distribution density in $w$. In particular, independent chi-squared distributions with $2\alpha$ and $2\beta$ degrees of freedom give the required ratio.

A compound mixed Poisson distribution is a random sum of independent claims whose count has a Poisson mixture law: conditional on a nonnegative random intensity $\Lambda$, $N$ has Poisson distribution with parameter $\Lambda$. The claim sizes are independent and identically distributed random variables, independent of the intensity and count mechanism. If $M_X$ is their moment-generating function, then $M_S(t)=\mathbb{E}\exp (\Lambda (M_X(t)-1))$ wherever finite. A gamma distribution intensity of shape $r$ and rate $\alpha$ gives a negative binomial distribution count with shape $r$ and success probability $\alpha /(\alpha +1)$. Independent intensities with common gamma rate add their shapes by additivity of independent gamma distributions with a common rate.

A nonnegative mean-reverting short-rate diffusion with positive parameters. Its mean-reversion level is $a/b$ and its diffusion size is proportional to $\sqrt{r}$. The Feller positivity condition for the CIR model controls whether zero is reachable. Its stationary law is a gamma distribution and its zero-coupon bond prices are exponential-affine.

For an independent statistical sample from the beta distribution with parameters $(\theta ,1)$, $\theta >0$, the probability density function is $\theta x^{\theta -1}$ on $(0,1)$. Its raw moments are $\mathbb{E}{X}^r=\theta /(\theta +r)$ for $r>-\theta$, by direct integration. Consequently the method of moments gives $T_1=\bar{X}/(1-\bar{X})$, whereas the maximum-likelihood estimator is $T_2=-n/{\sum }_i\log X_i$.

The second formula follows by setting the score function $n/\theta +{\sum }_i\log X_i$ to zero; its derivative is negative. Also $-\log X_i$ has an exponential distribution of rate $\theta$, since $\mathbb{P}(-\log X_i>u)=e^{-\theta u}$. The central limit theorem and delta method giveThus the asymptotic relative efficiency of $T_1$ relative to $T_2$ is $1-(\theta +1{)}^{-2}$. The transformed sum has a gamma distribution of shape $n$ and rate $\theta$, which also gives $\mathbb{E}{T}_2=n\theta /(n-1)$ for $n>1$ and $\mathrm{Var}(T_2)=n^2{\theta }^2/[(n-1{)}^2(n-2)]$ for $n>2$.

For independent exponential distribution observations with rate $\lambda$, the likelihood ratio for a larger simple rate is decreasing in their sum. The Neyman-Pearson lemma therefore selects a lower-tail sum threshold $c$. The sum has a gamma distribution, giving the displayed power; its derivative is $c{e}^{-\lambda c}(\lambda c{)}^{n-1}/(n-1)!>0$. Calibrating at ${\lambda }_0$ gives size $\alpha$ for the composite null $\lambda \le {\lambda }_0$ and a uniformly most powerful test against larger rates.

The probability density function of a Gamma distribution with shape $k>0$ and rate $\theta >0$. Integration against $y^{-j}$ gives $\mathbb{E}[Y^{-j}]={\theta }^j\Gamma (k-j)/\Gamma (k)$ when $j<k$; the integral diverges for $j\ge k$. Rate and scale are reciprocal conventions and must be distinguished.

Let a frailty random variable have a gamma distribution with mean one and variance $v>0$. Its shape and rate are both $1/v$. For conditional hazard function $U{a}^z{h}_0(t)$, $z\in \{0,1\}$, averaging the conditional survivor function gives $(1+v{a}^z{H}_0(t){)}^{-1/v}$. Differentiating its logarithm yields the population hazard function $a^z{h}_0(t)/(1+v{a}^z{H}_0(t))$ and hence the displayed hazard ratio. Writing $x=v{H}_0(t)$ gives $r-1=(a-1)/(1+ax)$. Therefore the population hazard ratio moves monotonically from $a$ towards one as exposure $x$ increases, even though the individual conditional hazard ratio remains $a$. The limit one requires $H_0(t)\to \infty$ for fixed positive $v$, or $v\to \infty$ for fixed positive $H_0(t)$; it is not a consequence of elapsed time alone. This is survival selection through preferential removal of larger frailties.

The Gamma function extends factorials to a continuous argument. Its defining Gamma integral iswhere $t^{z-1}=\exp ((z-1)\log t)$ uses the real natural logarithm for $t>0$. The Gamma function recurrence gives its meromorphic continuation to the complex plane, with simple poles at $0,-1,-2,\dots$ and no zeros. See Gamma function residue at a nonpositive integer and Gamma function has no zeros for proof outlines.^[ref]

For nonnegative integers $n$, the factorial identity has a shift of one:Thus $\Gamma (n)=(n-1)!$ for positive integers $n$. Half-integer arguments supply a different familiar constant:The first value follows from the Gaussian integral; the second follows from the Gamma function recurrence.^[ref]

The plot shows only positive real arguments; the gamma function also accepts complex numbers away from its poles. Matching the factorials alone does not select a unique interpolation, even if the Gamma function recurrence is imposed: $f(x)=\Gamma (x)\exp (\sin (2\pi x))$ has the same positive-integer values and the same recurrence relation. The Bohr-Mollerup theorem supplies the extra condition that selects the gamma function.

The useful idea is to replace a discrete product of numbers by an integral whose exponent varies continuously. The Gamma integral then evaluates whole families of integrals at once, provides the normalizing constant for the Gamma distribution in probability theory, and appears in physics, for example in the Bose integral. Its large-positive-argument growth reconnects this continuous description to factorials through Stirling's formula.^[ref]

Historically, Leonhard Euler studied the interpolation of factorials in his paper E19, written in 1729 and published in 1738. His integral over $(0,1)$ becomes the Gamma integral after a change of variables, with the same one-step shift in the argument.^[ref]

On $z>0$, consider $p_t+a{p}_z=D(z{p}_z{)}_z$ with $a,D>0$, zero scalar flux $ap-Dz{p}_z$ at the endpoints for $t>0$, and unit initial impulse at the origin. Put $r=a/D$ and $\eta =z/(Dt)$. The normalized similarity solution isThe ansatz $p=(Dt{)}^{-1}f(\eta )$ gives $[\eta f^{\prime }+(\eta -r)f{]}^{\prime }=0$. Endpoint decay sets this integrated constant to zero and hence $f\propto {\eta }^r{e}^{-\eta }$; the gamma function normalizes its integral to one. Its scale is $Dt$, so it converges weakly to a unit impulse as $t\downarrow 0$. Its gamma distribution shape is $1+r$, its maximum occurs at $z=at$, and its expected value is $(a+D)t$. For width proportional to $z$, the associated horizontally averaged plume concentration is proportional to $z^{r-1}{e}^{-z/(Dt)}$: its interior maximum is $(a-D)t$ when $a>D$, its supremum occurs at the origin when $a=D$, and it is singular there when $0<a<D$. The physical finite source regularizes this ideal-origin behaviour.

For real $a>0$ and $\mathrm{Re}s>0$, the change of variables $u=at$ in the defining improper integral for the Gamma function givesThe improper integral converges absolutely: near zero its absolute integrand behaves like $t^{\mathrm{Re}s-1}$, and at infinity the exponential function overcomes every fixed power.^[ref]

For real $s=1,2,4$ and $a=1$, the integrands are $e^{-t}$, $t{e}^{-t}$, and $t^3{e}^{-t}$, with total areas $\Gamma (1)=1$, $\Gamma (2)=1$, and $\Gamma (4)=6$. The third integrand is taller and has six times the area of the first, although the exponential function makes each tail decay.

For real $s>0$, the kernel $t^{s-1}{e}^{-t}$ becomes the probability density function of a Gamma distribution with shape $s$ and unit rate only after division by its normalizing constant $\Gamma (s)$. In particular, the shape-four probability density function is $t^3{e}^{-t}/6$. More generally, the Gamma integral verifies that $a^s{t}^{s-1}{e}^{-at}/\Gamma (s)$ integrates to one for shape $s>0$ and rate $a>0$.

For the half-integer evaluation, the change of variables $t=u^2$ reduces the Gamma integral to the Gaussian integral:Consequently, for real $u>0$,

For independent Poisson distributions with means $u{e}^{{\eta }_k}$ and a shape–rate Gamma distribution prior on $u$, integrating $u$ adds $a$ to the total-count exponent and $b$ to the sum of relative rates. This distinguishes gamma mixing from conditioning on the total count, which gives a multinomial distribution.

If $\Lambda$ has gamma distribution with shape $2$ and rate $p/q$, $N\mid \Lambda$ has Poisson distribution with intensity $\Lambda$, and the independent claim sizes have exponential distribution of mean $\mu$, where $p+q=1$, the aggregate law is $p^2{\delta }_0+2pq\mathrm{Exp}(p/\mu )+q^2\mathrm{Gamma}(2,\text{rate }p/\mu )$. This follows by expanding its moment-generating function as $[p+qp/(p-\mu t){]}^2$.

A Gamma–Poisson hierarchical model uses $Y_i\mid {\theta }_i\sim \mathrm{Poisson}(t_i{\theta }_i)$, ${\theta }_i\mid b\sim \mathrm{Gamma}(a,b)$, and $b\sim \mathrm{Gamma}(c,r)$, with gamma distributions in shape-rate convention. Conditional independence gives full conditional distributions ${\theta }_i\mid b,y\sim \mathrm{Gamma}(a+y_i,b+t_i)$ and $b\mid \theta ,y\sim \mathrm{Gamma}(c+na,r+{\sum }_i{\theta }_i)$.

For inverse-gamma distribution observations of known shape $k$ and scale $\Theta$, their likelihood function in $\Theta$ is proportional to ${\Theta }^{nk}\exp (-\Theta {\sum }_i{x}_i^{-1})$. A shape-$\alpha$, rate-$\lambda$ gamma distribution conjugate prior therefore gives the displayed Bayesian posterior. The Bayes estimator under squared error loss of the conditional claim expected value $\Theta /(k-1)$ is $(\alpha +nk)/[(k-1)(\lambda +{\sum }_i{x}_i^{-1})]$. It depends on reciprocals, rather than the sample mean of the claims.

Let the positive multiplier $U_i$ have a gamma distribution with mean $\tau$ and variance $\theta$, and conditionally let the repeated counts be independent with means $U_i{a}_{ij}$. Total expectation and variance giveThe Poisson-gamma mixture gives negative-binomial marginal counts of size ${\tau }^2/\theta$. Despite the shared multiplier, the marginal correlations generally depend on both marginal means and need not be exchangeable.