A lamp has three bulbs, each of a type with average lifetime 800 hours. If we model the probability of failure of a bulb by an exponential density function with mean 800, find the probability that all three bulbs fail within a total of 1000 hours.
To find: The probability of all the bulb fail within 1000 hours.
The lifetime of each bulb has an exponential density function and each bulb has an average lifetime of 800 hours.
The exponential density function is,
Let X, Y and Z be the independent random variable which denotes the three bulbs of the lamp. Therefore, the exponential density function of each bulb is,
The probability that all the bulbs fails within 1000 hours is denoted by, and the joint density function is,
Since, , y varies from 0 to and z varies from 0 to . Therefore, the value of is computed as follows.
Integrate with respect to z.
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