## a
lambda <- 1 / seq(1, 1000, by = 0.1)
plot(lambda, 1 - pexp(30, lambda), type = "l")
## b
ll <- function(lambda) {
(1 - pexp(30, lambda))^8 * dexp(6, lambda) * dexp(28, lambda)
}
lambda <- c(1e-5, seq(0, 0.04, by = 0.0001))
ml <- max(ll(lambda))
plot(lambda, ll(lambda) / ml, type = "l")
## c
optimise(ll, c(0, 0.05), maximum = TRUE)
## ll = 8 * log(1 - pexp(30, lambda)) + log(dexp(6, lambda)) +
## log(dexp(28, lambda))
## = 8 * 30 * lambda - 2 * log(lambda) + (6+28)*lambda
##
## ll' = 274 -2/lambda
## lambda.hat = 2/274
optimise(ll, c(0, 0.05), maximum = TRUE)$maximum - 1 / 137
## d)
qexp(0.9, 1 / 137)
## e)
ll.e <- function(lambda) {
(1 - pexp(30, lambda))^8 * pexp(20, lambda)^2
}
lambda <- c(1e-5, seq(0, 0.04, by = 0.0001))
ml <- max(ll.e(lambda))
lines(lambda, ll.e(lambda) / ml, type = "l", col = 2)
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