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Chapter 8: Q 8.7 (page 391)

A person has 100 light bulbs whose lifetimes are independent exponentials with mean 5 hours. If the bulbs are used one at a time, with a failed bulb being replaced immediately by a new one, approximate the probability that there is still a working bulb after 525 hours.

Short Answer

Expert verified

P{S100>525}=0.3085

Step by step solution

01

Step 1. Given information

Let Xibe a random variable that represents the lifetime of the ithbulb, i = 1,...,100.

02

Step 2. Mean and variance of Xi

Xiare i.i.d.'s all having exponential distribution mean 5 hours and variance 25 hours.

E(Xi)=5hoursV(Xi)=52=25hours

03

Step 3. Defining S100

S100=鈭�Xii=1100

04

Step 4. To find

P{S100>525}

05

Step 5. By using central limit theorem

P{S100>525}=PS100-100脳5100脳25>525-100脳5100脳25P{S100>525}=PS100-5002500>525-5002500P{S100>525}=PZ>2550P{S100>525}=PZ>0.5P{S100>525}=1-P{Z<0.5}=1-0.6915P{S100>525}=0.3085

06

Step 6. Final answer

The probability that there is still a working bulb after 525 hours is 0.3085

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