Question

5. The City of Rulertown consists of the unit interval [0, 1]. Rulertown needs to determine where to build the city's only fire station. It knows that the probability that a given fire occurs at location x, has the following probability density function (pdf): 1, if 0 5x 51 f(x) = o, otherwise (a) (5) Rulertown wants to minimize the average distance between the fire station and a fire. Where should the fire station be located? (b) (5) Where should the fire station be located if x has the following new pdf: fr) - 2x, if 0<xsi 0, otherwise

Answer 1

Answer Given Probability occurs at fire location X, has that a given the following probability density function (Pett) Z lo, if ocasi otherwise (a) Here F(x) = (a fe EK) = EC) = 3 [-] E(X) = [] E(X) = (1-0) E(X) = E(x) =0.5

③ Therefore the fire station should locate at x=05 (b) where we should the fire station be located if a has the following new pat f(x) = 2x, if osasi o otherwise Now E(x) = 1 x. 2x da E (x) = 2x² dx E(x) = 2 x²dx EM) = -e'] EM 2 E(X) -2 (1-0) F(x) = 24) E(x) = 0.6667 - 0.667 Therefore the fire station should locate at 0.667]