The lifetime (in weeks) of a certain system component is thought to have an exponential distribution, whose density func

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The lifetime (in weeks) of a certain system component is thought
to have an exponential
distribution, whose density function is given by
f(x) = λe−λx, x ≥ 0,
for some strictly positive parameter λ.
(a) The reliability of a component is defined as the probability
that the component will not have failed by a specified time. If the
reliability of the system component at 10.5 weeks is 0.9, find the
reliability at 10 weeks.
(b) One hundred components of this type are put in a new
system. All components that have failed are replaced at 20 week
intervals, and none are replaced at other times.
If R is the number of components that have to be replaced at the
end of the first interval, state assumptions and compute the mean
and variance of R.
Explain why this result holds for any such interval, and not
just the first.