Question

1. (20 points total) (a) (10 points) You have 100 light bulbs whose lifetimes are modeled by an indepen- dent exponential distribution with a mean of 8 hours. The bulbs are used one at a time, with a failed bulb being replaced immediately by a new one. i. (5 pointsUse the central limit theorem to approximate the probability that there is still a working bulb after 850 hours. ii. (5 points) Use the central limit theorem to approximate the probability that there is still a working bulb between 790 hours and 820 hours.

Answer 1

Xi ~ Exponertial le=8 T-8 sulubion: - X= x,+Xq+Xg... X100 a) lx= 100x 8 = 800 Tx = V1oox 8 = 80 P(x> 850) - P1z> 8.50-800 Plz>8586800) /2=X-49 = //23 0.62) に 0.2676 820-810) CEL p i) P17904 xc 890) - 790-soo 80 0.1465 = P(-0.12 L 220.25) 16-0 = = b) e=8 813 3 xin uniform vie 3 lx= 100*8 = soo o= V100 x 8 3 = 8003 D) P[x3 850) - P123850 - 800): 11231-08) So NB 0.140) i P ( 790 < xc 820) 20-609 a Cza 790- 800 8003 3 So V3 3 P(-0.22 LIL 0.43) 0.2535