Question

The​ life, in​ years, of a certain type of electrical switch has an exponential distribution with an average life β=2. If 300 of these switches are installed in different​ systems, what is the probability that at most 100 fail during the first​ year?

Answer 1

probability that a switch fail within 1 year =1-e^-x/=1-e^-1/2 =0.3935

n=

300

p=

0.3935

here mean of distribution=μ=np=		118.04
and standard deviation σ=sqrt(np(1-p))=		8.46
for normal distribution z score =(X-μ)/σx

therefore from normal approximation of binomial distribution and continuity correction:

probability that at most 100 fail during the first​ year :

probability =P(X<100.5)=(Z<(100.5-118.04080208621)/8.461)=P(Z<-2.07)=0.0192