Problem 183E

Consider the lifetime of a laser in Example 4-26. Determine the following in parts (a) and (b):

(a) Probability the lifetime is less than 1000 hours

(b) Probability the lifetime is less than 11,000 hours given that it is more than 10,000 hours

(c) Compare the answers to parts (a) and (b) and comment on any differences between the lognormal and exponential distributions.

Solution 183E

Step1 of 4:

Let us consider a random variable X it presents the lifetime of a Laser. And it follows lognormal distribution with parameters

Here our goal is:

a). We need to find the probability the lifetime is less than 1000 hours.

b). We need to find the probability the lifetime is less than 11,000 hours given that it is more than 10,000 hours.

c). We need to compare the answers to parts (a) and (b).

Step2 of 4:

a).

Suppose a random variable X follows lognormal distribution with parameters Then, The cumulative distribution of lognormal distribution is:

Now,

Where,is obtained from standard normal table(area under normal curve).

(In area under normal curve we have to see in row -2.0 under column 0.06)

Hence,

Therefore, P(X <1000) = 0.0197.

Step3 of 4:

b).

Consider,

…...(1)

Where,

Where,is obtained from standard normal table(area under normal curve).

(In area under normal curve we have to see in row -0.5 under column 0.03)

Hence,

Therefore, P(X >10000) = 0.7019.

Let,

Where, is obtained from standard normal table(area under normal curve).

(In area under normal curve we have to see in row -0.4 under column 0.06)

(In area under normal curve we have to see in row -0.5 under column 0.03)

Hence,

Therefore,

Now, Substitute these values in equation (1), Then we get,

Therefore,

Step4 of 4:

c).

We observe that the chance of laser working is less than 12000 hours given that its lifetime is more than 10000 hours is little higher than the chance of a laser working less than 1000 hours.

From conditional probability it is observed that lognormal probability depends on conditional events ,whereas in exponential distribution the conditional events does not depends on it. Therefore, The lifetime of product that degrades over time is often modded by lognormal

Random variable.