指数分布族

Equation
f(y| \theta, \phi)=\exp[\frac{y\theta-b(\theta)}{a(\phi)}+c(y,\phi)]
\theta:正準母数(標準母数)
\phi:ちらばり母数(局外母数)
b(\theta):正規化定数

指数分布族の期待値:b'(\theta)
指数分布族の分散:a(\phi)b''(\theta)

正規分布
f(y|u,\phi^{2})\\=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp[-\frac{1}{2} \frac{(y-u)^{2}}{\sigma^{2}}]\\=\exp[\frac{yu-u^{2}/2}{\sigma^{2}}-\frac{1}{2}(\frac{y^{2}}{\sigma^{2}}+\log_{e}(2\pi\sigma^{2}))]

2項分布
f(y|n,p)\\=\binom{n}{ny}p^{ny}(1-p)^{n-ny}\\=\exp[\frac{y\log_{e}(p/(1-p))-[-\log_{e}(1-p)]}{1/n}+\log_{e}\binom{n}{ny}]

R
20140708_1220140708_13

> y<-seq(-50,50,1)
> density<-dnorm(y,mean=0,sd=10)
> plot(y,density,type="l")
> 
> n<-100

> ny<-0:10

> p<-1/50

> probability<-dbinom(ny,size=n,prob=p)

> round(cbind(y=ny/n,ny,probability),5)
         y ny probability
 [1,] 0.00  0     0.13262
 [2,] 0.01  1     0.27065
 [3,] 0.02  2     0.27341
 [4,] 0.03  3     0.18228
 [5,] 0.04  4     0.09021
 [6,] 0.05  5     0.03535
 [7,] 0.06  6     0.01142
 [8,] 0.07  7     0.00313
 [9,] 0.08  8     0.00074
[10,] 0.09  9     0.00015
[11,] 0.10 10     0.00003

> plot(ny/n,probability,type="h")

> points(ny/n,probability,pch=1)

参考文献
服部環(2011).『心理・教育のためのRによるデータ解析』.福村出版.pp435

アプリケーション
R Core Team (2013). R: A language and environment for statistical computing.
R Foundation for Statistical Computing, Vienna, Austria.
URL http://www.R-project.org/.