Question

Problem 7: 8 points) The life X (in years) of a voltage regulator of a car has pdf f(x) Bane (9)* for x > 0. (a) Show that this is a valid p.d.f. [1] (b) Derive the c.d.f. F(x) of X [2] (C) Use your answer in (b) to find the probability that the regulator will last at least 7 years? [1] (d) Given that it has lasted at least 7 years, what is the conditional probability that it will last at least another 3.5 years? [2] (e) Find the median failure time and interpret it. [2]

Answer 1

7). x= The life (in years) of a voltage regulator of a car. pdf of x is xro f(x) = 3x2 a) Now, E P(x)=2 x> S f(x) dx = garander ()dre, -$*e*202 e-21.00 a Z 3) 3x = dz Total 1 So f(x) is a valid pdf b) edf of X.;F (1)= P6x3x)= 51.1) dt = faget 6 4 4 13 de e- as & zy *3/73 tla 110 * x3 =) 3t² dt = dy 73 x²/73 is early [-eisen 1- e-(317) c) probability that regulator will last at least 7 years = P6x3,7) = 1- P<x< 7) = 1- F(7):1-(1.01) - 0-3679 Given that it hon last at least 7 years for the conditional probability that it will last another 3.5 years - RB X 35 ** PGCX >10.5) nx37) = P(x>0.577)\x77) - P(*>7) P(*>10.5) IF(IOS) 1-(1-0-10:53) PC*37) IF(7) *te 0.03422 Chloei 30.093 1- e(1-0) 0.3679 e) us the median bek then, P(XSK) 20.5 >) F(K) 2 0.5 29 l-e-(K17)3 -0.5 7) e-(k/1): 0.5 -) -(): -0.69315 K= 2.33 so the median faliure time is 2.33 years.