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 a poisson discharged into the atmosphere from jet fuel. For a certain dose of the toxicant the study determines the survival time (in weeks) has a gamma distribution with λ= 1/10 and α= 5. What is the probability that a rat survives not more than 60 weeks?
Answers
Answer:
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)G(a,s) are: E(X)=s/aE(X)=s/a, Var(X)=as2Var(X)=as2, 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] 0.64
1.4 How to calculate P(X <= x) of a G?
We can calculate the cumulative distribution function at xx, FX(x)=P(X≤x)FX(x)=P(X≤x), as follows,
pgamma(1,shape=s,scale=a)## [1] 0.7134952# Empirical check p_emp <- mean(datos_G < 1) p_emp## [1] 0.71595
1.5 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] 0.103
2 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(μ,ϕ)IG(μ,ϕ) are: E(X)=μE(X)=μ, Var(X)=ϕμ3Var(X)=ϕμ3, 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)