Here is the problem discussed in class.

X stands for a random electronic component lifetime. The distribution X in hours is exponential, with rate parameter 1/1000 = .001. We assume that some system uses 8 of these components and that these 8 random lifetimes are mutually independent. We are going to simulate three situations of how the system works:

1. The system works as long as one component is still alive. (Max)
2. The system works only until the first failure occurs. (Min)
3. The system works as long as 6 components remain alive. (3rd order statistic)

Use R to randomly generate 10000 lifetimes and then 1000 iterates of component lifetimes for each of the 3 situations just described.

lifetimes <- rexp(10000,.001)
samples <- matrix(rexp(80000,.001),ncol=8,nrow=1000)
order.stats <- t(apply(samples,1,sort))
data1 <- order.stats[,8]
data2 <- order.stats[,1]
data3 <- order.stats[,3]

layout(matrix(1:4,2,2))
hist(lifetimes,prob=T,xlim=c(0,4500),main="Individual lifetimes")
hist(data1,prob=T,xlim=c(0,4500),main="Max")
hist(data2,prob=T,xlim=c(0,4500),main="Min")
hist(data3,prob=T,xlim=c(0,4500),main="Third Order Statistic")
layout(1)

xx <- seq(0,5000,len=201)
y1 <- .001*exp(-.001*xx)
y2 <- .008*exp(-.001*xx)*(1-exp(-.001*xx))^7
y3 <- .008*exp(-.008*xx)
y4 <- .168*exp(-.006*xx)*(1-exp(-.001*xx))^2


plot(xx,y1,type='l',ylim=c(0,.0025))  # original single components pdf
lines(xx,y2,type='l',lty=2)   #max pdf
lines(xx,y3,type='l',lty=3)   #min pdf
lines(xx,y4,type='l',lty=4)   #3rd OS