Survival function

3. Given the survival function: S(t) = exp(-t?) derive the probability density function and the hazard function. 4. Derive X(t) = f(t) S(t) using the definition of the hazard function and basic definition of conditional probability. 5. Derive S(t) = e-A(t) using the definition of the hazard function. 6. Given the hazard function: adta-1 A (t ) = 1 + Ata' derive the survival function and the probability density function. 7. Prove that if T has an arbitrary continuous distribution, the cumulative hazard of T, A(T), has an exponential distribution with unit parameter.1) In survival analysis and reliability theory, for a positive and continuous random variable T, the hazard rate function Mt) is related to the cdf by FT(t) = 1 ~ exp [w [ham] . Suppose that the life distribution of an item in years has the hazard rate function h(t) = t2 t > 0. What is the probability that a) the item survives to at least 2 years? b) the item survives to at least 2 years, given it has survived at least 1 year