Consider a group of patients who have been treated for an acute disease such as cancer, and let X be the number of years a person lives after receiving the treatment (the survival time). Under suitable conditions, the density function for X will be f (x) = ke-kx for some constant k.

(a) The survival function S(x) is the probability that a person chosen at random from the group of patients survives until at least time x. Explain why S(x) = 1 - F (x), where F (x) is the cumulative distribution function for X, and compute S(x).

(b) Suppose that the probability is .90 that a patient will survive at least 5 years [S(5) = .90]. Find the constant k in the exponential density function f (x).