Answer Happy

9. In this exercise, we will explore the consequences of
assuming that the survival times follow an exponential
distribution.
(a) Suppose that a survival time follows an Exp(λ) distribution,
so that its density function is f(t) = λ exp(−λt). Using the
relationships provided in Exercise 8, show that S(t) = exp(−λt).
(b) Now suppose that each of n independent survival times follows
an Exp(λ) distribution. Write out an expression for the likelihood
function
(c) Show that the maximum likelihood estimator for λ is
(d) Use your answer to (c) to derive an estimator of the mean
survival time. Hint: For (d), recall that the mean of an Exp(λ)
random variable is 1/λ.
Question 8 for reference:
η n λ= Σδι Συ Yi. i=1 i=1
8. Recall that the survival function S(t), the hazard function h(t), and the density function f(t) are defined in (11.2), (11.9), and (11.11), respectively. Furthermore, define F(t) = 1 – S(t). Show that the following relationships hold: f(t) = dF(t)/dt S(t) = exp • (-["h(u)du).