1.

r1 <- (1/8) / (1/20)
r2 <- r1 * ((1/8) / (1/25+1/25)) 
pi0 <- 1 / (1 + r1+r2)
pi1 <- pi0 * r1
pi2 <- pi0 * r2
pi <- c(pi0,pi1,pi2)
pi

# desired proportion is the rate of repair completions while in state 0,
# divided by the overall rate of repair completions

propRepsState0 <- 
   pi0 * (1/8) / (pi0 * (1/8) + pi1 * (1/8) )

print(propRepsState0)


2.


findProbsKLater <- function(hmmOut,k) 
{
   nstates <- hmmOut@nstates
   u <- getpars(hmmOut)
   p <- u[(nstates+1):(nstates + nstates^2)]
   p <- matrix(p,nrow=nstates,byrow=T)
   prodK <- diag(nstates)
   for (i in 1:k) {
      prodK <- prodK %*% p
   }
   samplePath <- hmmOut@posterior$state
   lastState <- samplePath[length(samplePath)]
   prodK[lastState,]
}

library(hmmr)
data(sp500)
cls <- sp500$Close
z <- hmm(cls,3)
print(findProbsKLater(z,2))

3.

From our book:

f_Y(s) = s f_X(s) / EX

F_Y(t) = integral f_Y(s) ds from 0 to t
       = integral s f_X(s) ds from 0 to t / EX

Using integration by parts, with u = s and dv = f_X(s) ds, the above
integral becomes

uv - integral(v du)

s F_X(s) eval from 0 to t - integral F_X(s) ds from 0 to t =
tF_X(t) - integral F_X(s) ds from 0 to t

FY <- function(FX,c,t,EX=NULL)  
{      
   if (is.null(EX)) {
      F1 <- function(s) 1 - FX(s)
      EX <- integrate(F1,0,c)$value
   }

   uv <- t * FX(t) / EX
   ivdu <- integrate(FX,0,t)$value / EX

   uv - ivdu

}

FXexample <- function(s) s^2  # density is 2t on (0,1)

print(FY(FXexample,1,0.5,2/3))  # should be 1/8
print(FY(FXexample,1,0.5))  # should be 1/8