# Question: In this problem we revisit the light bulb problem Recall

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?

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?

## Answer to relevant Questions

Consider the light bulb problem. Suppose we do not necessarily want to wait for the light bulb to burn out before we make a decision as to which type of bulb is being tested. Therefore, a modified experiment is proposed. The ...A random variable has a CDF given by Fx (x) = (1 –e –z) u (x). (a) Find Pr (X > 3). (b) Find Pr (X < 5| X > 3). (c) Find Pr (X > 6 | X > 3). (d) Find Pr (|X –5| < 4||X –6| > 2). Suppose a random variable X has a PDF which is nonzero only on the interval [ 0, 8 ) . That is, the random variable cannot take on negative values. Prove that A certain random variable has a characteristic function given by Find the PDF of this random variable. Derive an expression for the moment- generating function of a Rayleigh random variable whose PDF isPost your question