直感的理解
$x_1, x_2, …, x_k$ が独立にパラメータ $\lambda$ の指数分布 に従うとき、$x_1+x_2+…+x_k$はshape $k$ scale $\lambda$ のガンマ分布となる。
累積分布
$$F(x)=\int_0^x f(\varepsilon) d\varepsilon=\frac{\gamma(k,x/\lambda)}{\Gamma(k)}
=\frac{\int_0^{x/\lambda}t^{k-1}e^{-t}dt}{\int_0^\infty t^{k-1}e^{-t}dt}$$
密度関数
$$f(x)=x^{k-1}\left(\frac{1}{\lambda}\right)^k\frac{e^{-x/\lambda}}{\Gamma(k)}, { } x>0$$
期待値
$$E(x)=k\lambda$$
分散
$$Var(x)=k\lambda^2$$
分布の形を比較する
shape=$k$，rate=1/scale=$1/\lambda$
shape=$k$が大きくなると，分布はなだらかになる。：試行回数を増やすと，当然合計到達距離は伸びる
png("gamma01.png",height=300,width=400)
f02<-function(x) dgamma(x,shape=1,rate=1)
curve(f02,0.001,10,xlim=c(0,10),ylim=c(0,0.4),col=1)
f02<-function(x) dgamma(x,shape=1.5,rate=1)
curve(f02,0.001,10,col=2,add=T)
f02<-function(x) dgamma(x,shape=2,rate=1)
curve(f02,0.001,10,col=3,add=T)
f02<-function(x) dgamma(x,shape=3,rate=1)
curve(f02,0.001,10,col=4,add=T)
legend(6,0.4,c("shape=1,rate=1",
"shape=1.5,rate=1",
"shape=2,rate=1",
"shape=3,rate=1"),
cex=0.6,col=1:4,lty=rep(1,4))
dev.off()
rate=1/scale=$1/\lambda$が小さくなると，分布はなだらかになる ：１回の期待到達距離が伸びれば，当然到達距離は伸びる。
png("gamma02.png",height=300,width=400)
f02<-function(x) dgamma(x,shape=1,rate=1)
curve(f02,0.001,10,xlim=c(0,10),ylim=c(0,0.4),col=1)
f02<-function(x) dgamma(x,shape=1,rate=1/2)
curve(f02,0.001,10,col=2,add=T)
f02<-function(x) dgamma(x,shape=1,rate=1/3)
curve(f02,0.001,10,col=3,add=T)
f02<-function(x) dgamma(x,shape=1,rate=1/5)
curve(f02,0.001,10,col=4,add=T)
legend(6,0.4,c("shape=1,rate=1","shape=1,rate=1/2","shape=1,rate=1/3","shape=1,rate=1/5"),
cex=0.6,col=1:4,lty=rep(1,4))
dev.off()
png("gamma03.png",height=300,width=400)
f02<-function(x) dgamma(x,shape=2,rate=1)
curve(f02,0.001,10,xlim=c(0,10),ylim=c(0,0.4),col=1)
f02<-function(x) dgamma(x,shape=2,rate=1/2)
curve(f02,0.001,10,col=2,add=T)
f02<-function(x) dgamma(x,shape=2,rate=1/3)
curve(f02,0.001,10,col=3,add=T)
f02<-function(x) dgamma(x,shape=2,rate=1/5)
curve(f02,0.001,10,col=4,add=T)
legend(6,0.4,c("shape=2,rate=1",
"shape=2,rate=1/2",
"shape=2,rate=1/3",
"shape=2,rate=1/5"),
cex=0.6,col=1:4,lty=rep(1,4))
dev.off()
期待値（$k\lambda$）を1に保ったまま，shape=$k$を小さくする（rate=1/scale=$1/\lambda$を小さくする）： 一方で0近辺に集中しながら，分布の裾野は大きい方にどんどん広くなる。
f02<-function(x) dgamma(x,shape=5,rate=5)
curve(f02,0.001,10,xlim=c(0,4),ylim=c(0,1),col=1)
f02<-function(x) dgamma(x,shape=2,rate=2)
curve(f02,0.001,10,col=2,add=T)
f02<-function(x) dgamma(x,shape=1,rate=1)
curve(f02,0.001,10,col=3,add=T)
f02<-function(x) dgamma(x,shape=0.5,rate=0.5)
curve(f02,0.001,10,col=4,add=T)
f02<-function(x) dgamma(x,shape=0.1,rate=0.1)
curve(f02,0.001,10,col=5,add=T)
legend(2,1,c("shape=5,rate=5",
"shape=2,rate=2",
"shape=1,rate=1",
"shape=0.5,rate=0.5",
"shape=0.1,rate=0.1"),
cex=0.6,col=1:5,lty=rep(1,5))