Probability of failure from a particular cause. Suppose that a series system can fail from one of K independent competing causes with cumulative life distributions F,(y), . . . , F,( y j.

(a)    Derive an expression for the proportion PA( v) of the population that fail from cause k by age y.

(b)   Show P,(yj+ ... +P,(y)=F(y), the cumulative distribution of time to failure when all causes act.

(c)    Give the special result from (a) when the distributions are all exponential with means O,, k= I,. . ., K.

(d)   For (c), what proportion of the failures by age y have come from cause k?

(e)   Give the special result from (a) when all distributions are Weibull distributions with differing scale parameters and a common shape parameter

(f)     For (ej, what proportion of the failures by agey have come from cause k