If X 1 , , X n are independent exponentially distributed random variables, each with parameter , prove that the random variable whose value is given by the minimum of X 1 , , X n also has an exponential distribution, with parameter n In particular, if a complex piece of machinery consists of six pa...

Each exponential random variable has pdf and distribu ...

If X 1 , . . ., X n are independent exponentially distributed random variables, each with

Question:

If X₁, . . ., X_n are independent exponentially distributed random variables, each with parameter λ, prove that the random variable whose value is given by the minimum of {X₁, . . ., X_n} also has an exponential distribution, with parameter nλ. In particular, if a complex piece of machinery consists of six parts, each of which has an exponential distribution of time to failure with mean 2000 hours, and if the machine fails as soon as any of its parts fail, find the probability that the time to failure exceeds 300 hours.

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