At time t = 0, 19 identical components are tested. the lifetime distribution of each is exponential with parameter λ. the experimenter then leaves the test facility unmonitored. on his return 24 hours later, the experimenter immediately terminates the test after noticing that y = 13 of the 19 components are still in operation (so 6 have failed). derive the mle of λ. [hint: let y = the number that survive 24 hours. then y ~ bin(n, p). what is the mle of p? now notice that p = p(xi ≥ 24), where xi is exponentially distributed. this relates λ to p, so the former can be estimated once the latter has been.] (round your answer to four decimal places.)

Nthe opening scene of the movie raiders of the lost ark, indiana jones replaces a golden statue with a bag of sand. the platform on which the statue is placed is designed to detect the mass of the statue. if the bag of sand has a different mass than the statue, then a mechanism triggers and destruction ensues. the density of gold is about 20\text{ g/cm}^320 g/cm 3 20, space, g, slash, c, m, start superscript, 3, end superscript. the density of sand can be estimated at 2.5\text{ g/cm}^32.5 g/cm 3 2, point, 5, space, g, slash, c, m, start superscript, 3, end superscript. the gold statue can be modeled as a cylinder with height 25\text{ cm}25 cm25, space, c, m and radius 8\text{ cm}8 cm8, space, c, m. in terms of \piπpi, what is the mass of sand indiana jones needs to avoid triggering the mechanism?