Question

10. (10 points) One has 600 light bulbs whose life times are independent exponential random variables with parameter λ-1/10 hours. Find the approximate probability that there are at least 250 bulbs which last longer than 10 hours

Answer 1

here P(a bulb last longer than 10 hours)=P(X>10)=e^{- $\lambda$ t}=e^-(1/10)*10 =0.3679

hence expected number of bulbs last longer than 10 hours=np=600*0.3679=220.74

also std deviation =sqrt(np(1-p))=11.8123

for normal distribution z score =(X-mean)/std deviation

hence from normal approximation and continuity correction:

P(at least 250 bulbs last longer than 10 Hours)=P(X>=250)=1-P(X<=249)

=1-P(Z<(249.5-220.74)/11.81)=1-P(Z<2.43)=1-0.9925=0.0075