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Gamma Distribution

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A gamma distribution is a general type of statistical distribution that is related to the beta distribution and arises naturally in processes for which the waiting times between Poisson distributed events are relevant. Gamma distributions have two free parameters, labeled alpha and theta, a few of which are illustrated above.

Consider the distribution function D(x) of waiting times until the hth Poisson event given a Poisson distribution with a rate of change lambda,

D(x)=P(X<=x)
(1)
=1-P(X>x)
(2)
=1-sum_(k=0)^(h-1)((lambdax)^ke^(-lambdax))/(k!)
(3)
=1-e^(-lambdax)sum_(k=0)^(h-1)((lambdax)^k)/(k!)
(4)
=1-(Gamma(h,xlambda))/(Gamma(h))
(5)

for x in [0,infty), where Gamma(x) is a complete gamma function, and Gamma(a,x) an incomplete gamma function. With h an integer, this distribution is a special case known as the Erlang distribution.

The corresponding probability function P(x) of waiting times until the hth Poisson event is then obtained by differentiating D(x),

P(x)=D^'(x)
(6)
=lambdae^(-lambdax)sum_(k=0)^(h-1)((lambdax)^k)/(k!)-e^(-lambdax)sum_(k=0)^(h-1)(k(lambdax)^(k-1)lambda)/(k!)
(7)
=lambdae^(-lambdax)+lambdae^(-lambdax)sum_(k=1)^(h-1)((lambdax)^k)/(k!)-e^(-lambdax)sum_(k=1)^(h-1)(k(lambdax)^(k-1)lambda)/(k!)
(8)
=lambdae^(-lambdax)-lambdae^(-lambdax)sum_(k=1)^(h-1)[(k(lambdax)^(k-1))/(k!)-((lambdax)^k)/(k!)]
(9)
=lambdae^(-lambdax){1-sum_(k=1)^(h-1)[((lambdax)^(k-1))/((k-1)!)-((lambdax)^k)/(k!)]}
(10)
=lambdae^(-lambdax){1-[1-((lambdax)^(h-1))/((h-1)!)]}
(11)
=(lambda(lambdax)^(h-1))/((h-1)!)e^(-lambdax).
(12)

Now let alpha=h (not necessarily an integer) and define theta=1/lambda to be the time between changes. Then the above equation can be written

P(x)=(x^(alpha-1)e^(-x/theta))/(Gamma(alpha)theta^alpha)
(13)

for x in [0,infty). This is the probability function for the gamma distribution, and the corresponding distribution function is

D(x)=P(alpha,x/theta),
(14)

where P(a,z) is a regularized gamma function.

It is implemented in the Wolfram Language as the function GammaDistribution[alpha, theta].

The characteristic function describing this distribution is

phi(t)=F_x{(x^(-x/theta)x^(alpha-1))/(Gamma(alpha)theta^alpha)[1/2(1+sgnx)]}(t)
(15)
=(1-ittheta)^(-alpha),
(16)

where F_x[f](t) is the Fourier transform with parameters a=b=1, and the moment-generating function is

M(t)=int_0^infty(e^(tx)x^(alpha-1)e^(-x/theta)dx)/(Gamma(alpha)theta^alpha)
(17)
=int_0^infty(x^(alpha-1)e^(-(1-thetat)x/theta)dx)/(Gamma(alpha)theta^alpha).
(18)

giving moments about 0 of

mu_r^'=(theta^rGamma(alpha+r))/(Gamma(alpha))
(19)

(Papoulis 1984, p. 147).

In order to explicitly find the moments of the distribution using the moment-generating function, let

y=((1-thetat)x)/theta
(20)
dy=(1-thetat)/thetadx,
(21)

so

M(t)=int_0^infty((thetay)/(1-thetat))^(alpha-1)(e^(-y))/(Gamma(alpha)theta^alpha)(thetady)/(1-thetat)
(22)
=1/((1-thetat)^alphaGamma(alpha))int_0^inftyy^(alpha-1)e^(-y)dy
(23)
=1/((1-thetat)^alpha),
(24)

giving the logarithmic moment-generating function as

R(t)=-alphaln(1-thetat)
(25)
R^'(t)=(alphatheta)/(1-thetat)
(26)
R^('')(t)=(alphatheta^2)/((1-thetat)^2).
(27)

The mean, variance, skewness, and kurtosis are then

mu=alphatheta
(28)
sigma^2=alphatheta^2
(29)
gamma_1=2/(sqrt(alpha))
(30)
gamma_2=6/alpha.
(31)

The gamma distribution is closely related to other statistical distributions. If X_1, X_2, ..., X_n are independent random variates with a gamma distribution having parameters (alpha_1,theta), (alpha_2,theta), ..., (alpha_n,theta), then sum_(i=1)^(n)X_i is distributed as gamma with parameters

alpha=sum_(i=1)^(n)alpha_i
(32)
theta=theta.
(33)

Also, if X_1 and X_2 are independent random variates with a gamma distribution having parameters (alpha_1,theta) and (alpha_2,theta), then X_1/(X_1+X_2) is a beta distribution variate with parameters (alpha_1,alpha_2). Both can be derived as follows.

P(x_1,x_2)=1/(Gamma(alpha_1)Gamma(alpha_2))e^(x_1+x_2)x_1^(alpha_1-1)x_2^(alpha_2-1).
(34)

Let

u=x_1+x_2 x_1=uv
(35)
v=(x_1)/(x_1+x_2) x_2=u(1-v),
(36)

then the Jacobian is

J((x_1,x_2)/(u,v))=|v u; 1-v -u|=-u,
(37)

so

g(u,v)dudv=f(x,y)dxdy=f(x,y)ududv.
(38)
g(u,v)=u/(Gamma(alpha_1)Gamma(alpha_2))e^(-u)(uv)^(alpha_1-1)u^(alpha_2-1)(1-v)^(alpha_2-1)
(39)
=1/(Gamma(alpha_1)Gamma(alpha_2))e^(-u)u^(alpha_1+alpha_2-1)v^(alpha_1-1)(1-v)^(alpha_2-1).
(40)

The sum X_1+X_2 therefore has the distribution

f(u)=f(x_1+x_2)=int_0^1g(u,v)dv=(e^(-u)u^(alpha_1+alpha_2-1))/(Gamma(alpha_1+alpha_2)),
(41)

which is a gamma distribution, and the ratio X_1/(X_1+X_2) has the distribution

h(v)=h((x_1)/(x_1+x_2))
(42)
=int_0^inftyg(u,v)du
(43)
=(v^(alpha_1-1)(1-v)^(alpha_2-1))/(B(alpha_1,alpha_2)),
(44)

where B is the beta function, which is a beta distribution.

If X and Y are gamma variates with parameters alpha_1 and alpha_2, the X/Y is a variate with a beta prime distribution with parameters alpha_1 and alpha_2. Let

u=x+y v=x/y,
(45)

then the Jacobian is

J((u,v)/(x,y))=|1 1; 1/y -x/(y^2)|=-(x+y)/(y^2)=-((1+v)^2)/u,
(46)

so

dxdy=u/((1+v)^2)dudv
(47)
g(u,v)=1/(Gamma(alpha_1)Gamma(alpha_2))e^(-u)((uv)/(1+v))^(alpha_1-1)(u/(1+v))^(alpha_2-1)u/((1+v)^2)
(48)
=1/(Gamma(alpha_1)Gamma(alpha_2))e^(-u)u^(alpha_1+alpha_2-1)v^(alpha_1-1)(1+v)^(-alpha_1-alpha_2).
(49)

The ratio X/Y therefore has the distribution

h(v)=int_0^inftyg(u,v)du=(v^(alpha_1-1)(1+v)^(-alpha_1-alpha_2))/(B(alpha_1,alpha_2)),
(50)

which is a beta prime distribution with parameters (alpha_1,alpha_2).

The "standard form" of the gamma distribution is given by letting y=x/theta, so dy=dx/theta and

P(y)dy=(x^(alpha-1)e^(-x/theta))/(Gamma(alpha)theta^alpha)dx
(51)
=((thetay)^(alpha-1)e^(-y))/(Gamma(alpha)theta^alpha)(thetady)
(52)
=(y^(alpha-1)e^(-y))/(Gamma(alpha))dy,
(53)

so the moments about 0 are

nu_r=1/(Gamma(alpha))int_0^inftye^(-x)x^(alpha-1+r)dx
(54)
=(Gamma(alpha+r))/(Gamma(alpha))
(55)
=(alpha)_r,
(56)

where (alpha)_r is the Pochhammer symbol. The moments about mu=mu_1 are then

mu_1=alpha
(57)
mu_2=alpha
(58)
mu_3=2alpha
(59)
mu_4=3alpha^2+6alpha.
(60)

The moment-generating function is

M(t)=1/((1-t)^alpha),
(61)

and the cumulant-generating function is

K(t)=alphaln(1-t)=alpha(t+1/2t^2+1/3t^3+...),
(62)

so the cumulants are

kappa_r=alphaGamma(r).
(63)

If X is a normal variate with mean mu and standard deviation sigma, then

Y=((X-mu)^2)/(2sigma^2)
(64)

is a standard gamma variate with parameter alpha=1/2.

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