3.1. The Binomial and Related Distributions 1653.1.18.One way of estimating the number of fish in a lake is the followingcapture-
recapturesampling scheme. Suppose there areN fish in the lake whereN is
unknown. A specified number of fishTare captured, tagged, and released back to
the lake. Then at a specified time and for a specified positive integerr,fishare
captured until therthtagged fish is caught. The random variable of interest isY
the number of nontagged fish caught.(a)What is the distribution ofY? Identify all parameters.(b)What isE(Y)andtheVar(Y)?(c)The method of moment estimate ofNis to setYequal to the expression for
E(Y) and solve this equation forN. Call the solutionNˆ. DetermineNˆ.(d)Determine the mean and variance ofNˆ.3.1.19.Consider a multinomial trial with outcomes 1, 2 ,...,kand respective prob-
abilitiesp 1 ,p 2 ,...,pk.Letpsdenote the R vector for (p 1 ,p 2 ,...,pk). Then a single
random trial of this multinomial is computed with the commandmultitrial(ps),
where the required R functions are:^2
psum <- function(v){
p<-0; psum <- c()
for(j in 1:length(v)){p<-p+v[j]; psum <- c(psum,p)}
return(psum)}
multitrial <- function(p){
pr <- c(0,psum(p))
r <- runif(1); ic <- 0; j <- 1
while(ic==0){if((r > pr[j]) && (r <= pr[j+1]))
{multitrial <-j; ic<-1}; j<- j+1}
return(multitrial)}(a)Compute 10 random trials ifps=c(.3,.2,.2,.2,.1).(b)Compute 10,000 random trials forpsas in (a). Check to see how close the
estimates ofpiare withpi.3.1.20.Using the experiment in part (a) of Exercise 3.1.19, consider a game when
a person pays $5 to play. If the trial results in a 1 or 2, she receives nothing; if a
3, she receives $1; if a 4, she receives $2; and if a 5, she receives $20. LetGbe her
gain.(a)DetermineE(G).(b)Write R code that simulates the gain. Then simulate it 10,000 times, collecting
the gains. Compute the average of these 10,000 gains and compare it with
E(G).(^2) Downloadable at the site listed in the Preface