Prove that each interrelationship for s(x), f(x) and h(x) as stated here are true with detailed steps for each of them.

Survival Analysis

the result that the mean residual life is related to the area under the survival curve is obtained by using a partial summation formula. 2. Interrelationships between the various quantities discussed earlier, for a continuous lifetime X, may be summarized as Prave S(x) = [ f(odt - exp(-/ PE- medienos du mrlu) = exp[-H(x)] mrl(0) exp mrl(x) d -S(x) dx b(x)S(x) f(x) mrl(0) mrl(x)2 exp du mrlu) b(x): mrl(x) + 1 dx d In[S(x)] dx f(x) S(x) 1) /mrl(x) mrl(x) mrl(x) + dx SS(u)du S(x) S*(u - x)/(u)du S(x) The Hazard Function A basic quantity, fundamental in survival analysis, is the hazard function. This function is also known as the conditional failure rate in reliability, the force of mortality in demography, the intensity function in stochastic processes, the age-specific failure rate in epidemiology, the inverse of the Mill's ratio in economics, or simply as the hazard rate. The hazard rate is defined by b(x) Plx X). It can be shown (see Theoretical Note 1) that the mean residual life is the area under the survival curve to the right of x divided by S(x). Note that the mean life, al = mrlo, is the total area under the survival curve. For a continuous random variable, T-x)Odi S sodi mrl(x2) (2.4.1) S(x) S(x) and = E(X) = E irodi = ["S s() dr. (2.4.2) Also the variance of X is related to the survival function by Var(x) = 2 X). It can be shown (see Theoretical Note 1) that the mean residual life is the area under the survival curve to the right of x divided by S(x). Note that the mean life, al = mrlo, is the total area under the survival curve. For a continuous random variable, T-x)Odi S sodi mrl(x2) (2.4.1) S(x) S(x) and = E(X) = E irodi = ["S s() dr. (2.4.2) Also the variance of X is related to the survival function by Var(x) = 2