Question

Problem 7: (8 points) The length of time to failure (in hundred of hours) for a transistor is a random variable Y with e.df. given by Fy) - 1 - exp{-1}, if y> 0 0, elsewhere. (a) Find the p.d.ff(s) of Y and show that it is indeed a valid p.d.f[2] (b) Find the 30 percentile of Y and interpret it (2) (c) Find E(Y) and V(Y) (2) (d) Find the probability that the transistor operates for at least 200 hours. [1] (e) Find P(100 <Y < 200). [1]

Answer 1

Coven that if yo theswise F192 sl- exp(-42) The pdf of 7 is fryd Fy Jy 29 exPC-8° ) = f(y) = 4 احمر 2 if yzo Thesurse figis pot of 7 then it is satisfy the Conditions Sy fly, dy = 1 So zy eye dy dt where to y² - et 2 2y eye dy =) fly) is the pdf off Proved ba 30th percentile is 0.30 Plys y) = Fly) = Przy) 1- exp(-y?) = 09:30 - y2 log (6.70) =_=0:35 67 y = 0.5972 -2

classmate Page The length of time to railue upto hudred of hows) with propability 0.5972 (in 0:30. car Elya Jy y fly ) dy 2 9* e-82 let y² = t 27 dy - dt Le te e at det 3/21 وله { ایسا 10.8862 = E(4) = 0.8862 Elwy2 = I y² fly dy 2 Let y² = t zydy a dt 12 - Lot et at a E() - (E(%))² = Valy) - 1-(0-8862 ² = 0:2146 (varly)= 0.2146 fry P(12) As + PlyZ2) exp(-4) O 0183 CS Probability that the operaters for at least 200 hours. CamScannfi P1472) = 0:0183] Anx