In this problem, we revisit the light bulb. Recall that there were two types of bulbs, long- life (L) and short- life (S) and we were given a box of unmarked bulbs and needed to identify which type of bulbs are in the box. In Exercise 3.43, we chose to run one of the bulbs until it burned out in order to help us identify which type of bulbs are in the box. This time, in order to obtain a more reliable decision, we are going to burn two different bulbs from the box, observe how long it takes each bulb to burn out, and then make a decision as to what type of bulbs are in the box. Let X represent the time that it takes the first bulb to burn out and let Y represent the time it takes the second bulb to burn out. It would seem reasonable to assume that X and Y are independent and since both bulbs are taken from the same box, the PDFs of there lifetimes should be the same. Modeling the conditional PDFs as in Exercise 3.43, we have
The a priori probability of the bulb types were Pr (S) = 0.75 and Pr (L) = 0.25.
(a) If the two bulbs are tested and it is observed that the first bulb burns out after 200 h and the second bulb burns out after 75 h, which type of bulb was most likely tested?
(b) What is the probability that your decision in part (b) was incorrect?
(c) Determine what decision should be made for each possible observation pair, {X = x, Y =y}.That is, divide the first quadrant of the (x, y)- plane into two regions, one including all sets of points for which we would decide that the bulbs are S- type and its complement where we decide the bulbs are L- type.

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