A system is built using two independent components, X and Y. The system designer may connect the components either in series (the system works only if both X and Y work) or in parallel (the system works if at least one of X or Y works). Assume P(X)=r1 and P(Y)=r2, where 0<r1,r2<1.
a. (i) Define, in the context of reliability analysis, what is meant by a series system and a parallel system.
b. (ii) Derive an expression for the reliability of the series system in terms of r1 and r2.
c. (iii) Derive an expression for the reliability of the parallel system in terms of r1 and r2.
d. (iv) Compare the two expressions obtained in parts (ii) and (iii) and explain why, for any 0<r1,r2<1, the parallel system is at least as reliable as each individual component, whereas the series system is at most as reliable as the less reliable component.