Gamma 分配

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程式設計與統計軟體報告
伽瑪分配
(The Gamma Probability
Distribution)
金融二 徐以軒
金融二 楊士鴻
The Gamma Probability
Distribution
 A random variable Y is said to have a gamma
distribution with parameters α>0 and β>0 if and only
if the density function of Y is
Gamma Function
Mean and Variance
Mean and Variance
shape=α表形狀參數;scale=s=β表尺度參數
Gamma Density Functions,
β=1
R Command
op=par(mfrow=c(2,2))用來合併多個圖表成為一張圖
curve(dgamma(x, 1,1), 0, 10)
curve(dgamma(x, 2,1), 0, 10)
curve(dgamma(x, 3,1), 0, 10)
curve(dgamma(x, 4,1), 0, 10)
Gamma Density Functions,
α=1
R Command
op=par(mfrow=c(2,2))
curve(dgamma(x, 1,1), 0, 10)
curve(dgamma(x, 1,2), 0, 10)
curve(dgamma(x, 1,3), 0, 10)
curve(dgamma(x, 1,4), 0, 10)
Chi-Square Distribution
 Let v be a positive integer. A random variable Y is said
to have a chi-square distribution with v degrees of
freedom if and only if Y is a gamma-distributed
random variable with parameters α=v/2 and β=2
Exponential Distribution
 A random variable Y is said to have an exponential
distribution with parameter β>0, α=1 if only if the
density function of Y is
In R Language
In R Language
Example
The magnitude of earthquakes recorded in a region of
Norh America can be modeled as having an exponetial
distribution with mean 2.4, as measured on the Richter
scale. Find the probability that an earthquake striking
the region will
(a) exceed 3.0 on the Richter scale
(b) fall between 2.0 and 3.0 on the Richter scale
Answer for Example
(a)
1-pgamma(3,1,1/2.4)
(b)
1-pgamma(2,1,1/2.4)-(1-pgamma(3,1,1/2.4))
References
 陳景祥(2014) 。《R軟體應用統計方法 》 。東華
 http://en.wikipedia.org/wiki/Gamma_distribution#Char
acterization_using_shape_k_and_scale_.CE.B8
(Wikipedia)
 http://ccckmit.wikidot.com/st:gamma (陳鍾誠)
 http://stat.ethz.ch/R-manual/Rpatched/library/stats/html/GammaDist.html (R
Language)
 Dennis D. Wacherly, William Mendehall III, Richard L.
Scheaffer, Mathematical Staticstics with Applications.
Practice
The operator of a pumping station has observed that
demand for water during early afternoon hours an
approximately exponential distribution with mean 100
cfs(cubic feet per second).
(a) Find the probability that the demand will exceed 200
cfs during the early afternoon on a randomly selected
day
(b) What water-pumping capacity should the station
maintain during early afternoon so that the probability
that demand will exceed capacity on a randomly
selected day is only .01?

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