The distribution of downtime T for breakdowns of a computer system is given by

where a is a positive constant. The cost of downtime derived from the disruption resulting from breakdowns rises exponentially with T: cost factor = h(T) = e^bT Show that the expected cost factor for downtime is [a/(a – b)]², provided that a > b.

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fr(t) = [ate-a (t> 0) 0 (otherwise)