Exponential Family Distributions
Carlos Alberto Cardozo Delgado
February 2020
In the following some of the most important members of the Exponential Family distributions.
- Normal Distribution
- Gamma Distribution
- Inverse Gaussian Distribution
- Binomial Distribution
- Poisson Distribution
- Negative Binomial Distribution
The next subsections introduce one example per distribution and how to calculate relevant quantities about them in R.
1 Gamma Distribution, G.
This is a typical situation where a G distribution is useful:
1.1 Initial Situation
In a biomedical study with rats, a dose-response investigation is used to determine the effect of the dose of a toxicant on their survival time. The toxicant is one that is frequently discharged into the atmosphere from jet fuel. For a certain dose of the toxicant, the study determines that the survival time, in weeks, has a Gamma distribution with a = 5 and s = 10. What is the probability that a rat survives no longer than 60 weeks? Walpole et. al (2012), pag. 198.
The mean and variance expressions of a \(G(a,s)\) are: \(E(X) = s/a\), \(Var(X) = as^2\), respectively.
1.2 Density funtion of a G
Some examples, with different parameters values of the density function of a G distribution:
soporte <- seq(0.01,10,length=50)
s <- 1
a <- 0.8
valores1_fd <- dgamma(soporte,shape=s,scale=a)
plot(soporte,valores1_fd,type='l',main="Gamma Densities")
a2 <- 3
valores2_fd <- dgamma(soporte,shape=s,scale=a2)
points(soporte,valores2_fd,type='l',col=2)
s2 <- 2
valores3_fd <- dgamma(soporte,shape=s2,scale=a)
points(soporte,valores3_fd,type='l',col=3)
1.3 Generating a random sample of a G
In this subsection we illustrate how to generate a random sample of a G distribution.
datos_G <- rgamma(20000,shape=s,scale=a)
hist(datos_G,main="Histogram of Gamma data")
# True mean
MT <- s*a
MT## [1] 0.8# Sample mean
mean(datos_G)## [1] 0.797204# True variance
VART <- a*(s^2)
VART## [1] 0.8# Sample variance
round(var(datos_G),2)## [1] 0.641.4 How to calculate P(X <= x) of a G?
We can calculate the cumulative distribution function at \(x\), \(F_X(x) =P(X \leq x)\), as follows,
pgamma(1,shape=s,scale=a)## [1] 0.7134952# Empirical check
p_emp <- mean(datos_G < 1)
p_emp## [1] 0.715951.5 How to obtain a quantile of a G?
This is an example of how to obtain the r quantile of a G distribution:
q_10 <- qgamma(0.1,shape=s,scale=a)
q_10## [1] 0.08428841# Empirical check
r_emp <- mean(datos_G <= q_10)
r_emp## [1] 0.1032 Inverse Gaussian Distribution, IG.
2.1 Initial Situations
The IG has been useful modeling failure times for air conditioning in Boeing 720 jets and levels of precipitation in storms, D. J. Best (2012) et. al. Advances of Decision Sciences, doi:10.1155/2012/150303.
The mean and variance expressions of a \(IG(\mu,\phi)\) are: \(E(X) = \mu\), \(Var(X) = \phi\mu^3\), respectively.
2.2 Density funtion of a IG
Some examples, with different parameters values of the density function of a IG distribution:
# In this case we are using the package "statmod"
require(statmod)## Loading required package: statmod## Warning: package 'statmod' was built under R version 3.6.3soporte <- seq(0.01,6,length=200)
mu <- 1
phi <- 0.8
valores1_fd <- dinvgauss(soporte,mean=mu,dispersion=phi)
plot(soporte,valores1_fd,type='l',main="Inverse Gaussian Densities ",ylim=c(0,1.5))
mu2 <- 3
valores2_fd <- dinvgauss(soporte,mean=mu2,dispersion=phi)
points(soporte,valores2_fd,type='l',col=2)
phi2 <- 2
valores3_fd <- dinvgauss(soporte,mean=mu,dispersion=phi2)
points(soporte,valores3_fd,type='l',col=3)
2.3 Generating a random sample of a IG
In this subsection we illustrate how to generate a random sample of a IG distribution.
datos_IG <- rinvgauss(20000,mean=mu,dispersion=phi)
hist(datos_IG,main="Histogram of Inverse Gaussian Data")
# True mean
MT <- mu
MT## [1] 1# Sample mean
mean(datos_IG)## [1] 1.008163# True variance
VART <- phi*(mu^3)
VART## [1] 0.8# Sample variance
round(var(datos_IG),3)## [1] 0.8252.4 How to calculate P(X <= x) of a IG?
We can calculate the cumulative distribution function at \(x\), \(F_X(x) =P(X \leq x)\), as follows,
pinvgauss(1,mean=mu,dispersion=phi)## [1] 0.6543968# Empirical check
p_emp <- mean(datos_IG < 1)
p_emp## [1] 0.65462.5 How to obtain a quantile of a IG?
This is an example of how to obtain the r quantile of a IG distribution:
r <- 0.1
q_10 <- qinvgauss(r,mean=mu,dispersion=phi)
q_10## [1] 0.2738706# Empirical check
r_emp <- mean(datos_IG <= q_10)
round(r_emp,3)## [1] 0.13 Binomial Distribution, B.
This is a typical situation where a B distribution is useful:
3.1 Initial Situation
The probability that a patient recovers from a rare blood disease is 0.4. If 15 people are known to have contracted this disease, what is the probability that
a- at least 10 survive
b- from 3 to 8 survive, and
c- exactly 5 survive? Walpole et. al (2012), pag. 146.
The mean and variance expressions of a \(B(p,n)\) are: \(E(X) = np\), \(Var(X) = n(1-p)p\), respectively.
3.2 Density funtion of a B
Some examples, with different parameters values of the density function of a B distribution:
p <- 0.2
size <- 20
par(mfrow = c(1,3))
soporte <- seq(0,size)
valores1_fd <- dbinom(soporte,prob=p,size=size)
plot(soporte,valores1_fd,type='h',ylab= " ",main="B Densities")
p2 <- 0.4
valores2_fd <- dbinom(soporte,prob=p2,size=size)
plot(soporte,valores2_fd,type='h',col=2)
p3 <- 0.7
valores3_fd <- dbinom(soporte,prob=p3,size=size)
plot(soporte,valores3_fd,type='h',col=3)
3.3 Generating a random sample of a B
In this subsection we illustrate how to generate a random sample of a B distribution.
set.seed(2020)
datos_B <- rbinom(10000,prob=p,size=size)
hist(datos_B,main="Histogram of Binomial Data")
# True mean
MT <- p*size
MT## [1] 4# Sample mean
mean(datos_B)## [1] 3.9787# True variance
VART <- p*(1-p)*size
VART## [1] 3.2# Sample variance
round(var(datos_B),2)## [1] 3.183.4 How to calculate P(X <= x) of a B?
We can calculate the cumulative distribution function at \(x\), \(F_X(x) =P(X \leq x)\), as follows,
x <- 1
pbinom(x,prob=p,size=size)## [1] 0.06917529# Empirical probability
p_emp <- mean(datos_B <= x)
p_emp## [1] 0.07163.5 How to obtain a quantile of a B?
This is an example of how to obtain the r quantile of a B distribution:
r <- 0.5
q_50 <- qbinom(r,prob=p,size=size,lower.tail=TRUE)
q_50## [1] 4# Empirical check
r_emp <- mean(datos_B <= q_50)
r_emp## [1] 0.63434 Poisson Distribution, P.
This is a typical situation where a P distribution is useful:
4.1 Initial Situation
Hospital administrators in large cities anguish about traffic in emergency rooms. At a particular hospital in a large city, the staff on hand cannot accommodate the patient traffic if there are more than 10 emergency cases in a given hour. It is assumed that patient arrival follows a Poisson process, and historical data suggest that, on the average, 5 emergencies arrive per hour.
a- What is the probability that in a given hour the staff cannot accommodate the patient traffic?
b- What is the probability that more than 20 emergencies arrive during a 3 hour shift?. Walpole et. al \((2012)\), pag. 166.
The mean and variance expressions of a \(P(\lambda)\) are: \(E(X) = \lambda\), \(Var(X) = \lambda\), respectively.
4.2 Density funtion of a P
Some examples, with different parameters values of the density function of a P distribution:
soporte <- seq(0,30)
lambda <- 1
par(mfrow = c(1,3))
valores1_fd <- dpois(soporte,lambda)
plot(soporte,valores1_fd,type='h',main="P Densities")
lambda2 <- 9
valores2_fd <- dpois(soporte,lambda2)
plot(soporte,valores2_fd,type='h',col=2)
lambda3 <- 17
valores3_fd <- dpois(soporte,lambda3)
plot(soporte,valores3_fd,type='h',col=3)
4.3 Generating a random sample of a P
In this subsection we illustrate how to generate a random sample of a P distribution.
lambda <- 9
datos_P <- rpois(10000,lambda)
hist(datos_P,main="Histogram of Poisson Data")
# True mean
lambda## [1] 9# Sample mean
mean(datos_P)## [1] 9.0176# True variance
lambda## [1] 9# Sample variance
round(var(datos_P),2)## [1] 8.874.4 How to calculate P(X <= x) of a P?
We can calculate the cumulative distribution function at \(x\), \(F_X(x) =P(X \leq x)\), as follows,
x <- 3
round(ppois(x,lambda),2)## [1] 0.02# Empirical probability
p_emp <- mean(datos_P <= x)
p_emp## [1] 0.01824.5 How to obtain a quantile of a P?
This is an example of how to obtain the r quantile of a P distribution:
r <- 0.4
q_40 <- qpois(r,9,lower.tail=TRUE)
q_40## [1] 8# Empirical Check
r_emp <- mean(datos_P <= q_40)
r_emp## [1] 0.45465 Negative Binomial Distribution, NB.
This is a typical situation where a NB distribution is useful:
5.1 Initial Situation
Consider the use of a drug that is known to be effective in \(60%\) of the cases where it is used. The drug will be considered a success if it is effective in bringing some degree of relief to the patient. We are interested in finding the probability that the fifth patient to experience relief is the seventh patient to receive the drug during a given week. Walpole et. al \((2012)\), pag. 158.
The mean and variance expressions of a \(NB(k,p)\) are: \(E(X) = k(1-p)/p\), \(Var(X) = k(1-p)/p^2\), respectively.
5.2 Density funtion of a NB
Some examples, with different parameters values of the density function of a NB distribution:
soporte <- seq(0,25)
p <- 0.75
k <- 2
par(mfrow = c(2,2))
valores1_fd <- dnbinom(soporte,prob=p,size=k)
plot(soporte,valores1_fd,type='h',main="NB Density")
p2 <- 0.5
valores2_fd <- dnbinom(soporte,prob=p2,size=k)
plot(soporte,valores2_fd,type='h',col=2)
k <- 5
valores3_fd <- dnbinom(soporte,prob=p2,size=k)
plot(soporte,valores3_fd,type='h',col=3)
k <- 10
valores4_fd <- dnbinom(soporte,prob=p2,size=k)
plot(soporte,valores4_fd,type='h',col=4)
5.3 Generating a random sample of a NB
In this subsection we illustrate how to generate a random sample of a NB distribution.
# Number of success
p <- 0.75
size <- 5
datos_NB <- rnbinom(10000,prob=p,size=size)
datos_NB[1:20]## [1] 2 1 3 4 2 7 1 4 2 2 1 0 2 0 0 3 3 4 1 2hist(datos_NB,main="Histogram of Negative Binomial Data")
# True mean
MT <- size*(1-p)/p
MT## [1] 1.666667# Sample mean
mean(datos_NB)## [1] 1.6618# True variance
VART <- size*(1-p)/p^2
VART## [1] 2.222222# Sample variance
round(var(datos_NB),2)## [1] 2.215.4 How to calculate P(X <= x) of a NB?
We can calculate the cumulative distribution function at \(x\), \(F_X(x) =P(X \leq x)\), as follows,
x <- 5
size <- 5
round(pnbinom(x,p=p,size=size),4)## [1] 0.9803# Empirical probability
p_emp <- mean(datos_NB <= x)
p_emp## [1] 0.98015.5 How to obtain a quantile of a NB?
This is an example of how to obtain the r quantile of a NB distribution:
r <- 0.5
size <- 5
q_50 <- qnbinom(r,prob=p,size=size,lower.tail=TRUE)
q_50## [1] 1# Empirical Check
r_emp <- mean(datos_NB <= q_50)
r_emp## [1] 0.534