Question

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

Answer 1

(in hundred howy variable y with c.df of transistor is а random a 7) The length of time to failure given by Fly) = 2 - - expl-y) >o elsewhere is (a) The pelf of fly) d(Fly) 2 + dy elsewhere Illi- exp(-y2) if y zo dy dlo) dy (-2y) expl-y²), if yo 0 elsewhere [ay expl-y²) if yzo O elsewhere

have to a Go show forove valid pdf, we fly) is ffly) dy 2 오 4 Now, y you (fly, dy z dye ye dy yzo het 27 y 2 ) dz = z) dz z dy din as → Zso and us y sao za Zo e-dz Z azo 720 Z 720 z eo) D) -le-co -10 I a valid z Thus, fly) is felf

To find the 30th 6 bercentile of Y K be the 30th percentile of y het then, 2 2 0.30 > 0.30 P[Y<k y=k >> fly) dy yao yok >> dye ydy ?о Zz y 2 » Zzk? e-zdz 20.30 het dzz dy dy Z+0. as y ok? 2) 20.30 zao e-2 Zak? =) z -1 220 e) -1(e-k? 2) (e- eo) 20.30 z 0.30 1 -0.30 e-k? ) 2 ork?

-ke o =) 0.7 log 10.7). > ak? 2 2) 0.1549 K? K z-ke 0.1549 0.3936 2 z Thuse The 30th bencentile the data is of 0.39.36

z Ely] fy.fly) dy 420 2 ly gerdy Note: we have already yo find in previous exerdse 2 Sayery'dy z let) lyžeykdy y zo 20 y-o y zoo z fyeye y layey'dy Holly / dy e Y dy) dy ((s. e. gdy yeye fe-y'dy -I yzo yuno + y zoo (0-0) zo 20 for tim yey? y soo lim z lim 1 Cby a-He rule) ysoo Qy ey² y sa 2

Thus, 20 0 05 Ery] = (- yeye fet'dy to Fey'dy fey'dy O t 0 ਹੈ0 2 ਹੈ Now, o ) •u du z ਕ ( using result) and f e-42 dee 2 ਕੇ & Ely] Z R

Elyly? yun fy²flysdy yo y zoo Jy² ay eye dy yo yooo N Tydery" aylly 2 Y2o het 22 y² ) dz z dydy 20_as y se 2000 as y so 2 I ze zdz 2=0 720 z (by partition rule of integertion ) (diz) (e-zdz dz dz 220 2 2e-3 fle. e) dz 720 Zza ze 7 + e-z N Exe? le

zzo |-२९-२ Zz0 Z lim (-1e2 - e-2) -२९- - e-- 2500 024 2 2) - (०.०० - CO.९० - ९०) lim-z 70 lim -L 2-50 -1.etz e-३ 2 -(0-1) 2 1 -0 + 2 0 + । 2 (८५) : E[y2] - (5) (" 1 - 2 1 2 - Ro Z 4 Linn

probability that the transistor operates 000 hours is a The Sobability for at least P[*>200] fly) dy z doo z het Z= y ² z dz z dy dey Sayerst day Te zdz as y $200 35 754.00 =) Zao 2200 yosoo E 27400 290 z 22400 z ୧ -( - co z -400 € 400) e-400 1.915 x 10144 2 2

P (100 54 (200) y=200 z f fly, dy yzdDO 2 42,00 (dye-y²dy Je too 22(200) 2 e 7 de 721100) 2 22600) het Zay² dz z dydy 2 X-> (100) og za) (200) & as ya) 1 • 100 28 y 200 z -1 72 110o)² Z -le-40000 10000 e - 40000 - 10000 z le