In this problem, we revisit the light bulb problem. Recall that there were two types of light bulbs, long- life ( L) and short- life ( S) and we were given an unmarked bulb and needed to identify which type of bulb it was by observing how long it functioned before it burned out. Suppose we modify the problem so that the lifetime of the bulbs is modeled with a continuous random variable. In particular, suppose the two conditional PDFs are now given by

Where is the random variable that measures the lifetime of the bulb in hours. The a priori probabilities of the bulb type were Pr (s) = 0.75 and Pr (L) =0.25.
(a) If a bulb is tested and it is observed that the bulb burns out after 200 h, which type of bulb was most likely tested?
(b) What is the probability that your decision in part (a) was incorrect?

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= 100 exp(-100)-(x) and fx12(x)-T10 exp(T000)-(x) /rls(x) --expl__ |u(x) and )11 x)