The time till failure of an electronic component has an Expo

The time till failure of an electronic component has an Exponential distribution and it is known that 10% of components have failed by 1000 hours.

(a) What is the probability that a component is still working after 5000 hours?

(b) Find the mean and standard deviation of the time till failure.

I am not sure what to use as values for lambda and X. If lambda is the \"rate parameter\", would it be the complement of the percentage that fails(10%), so 0.9 or 9/10?

Are the bounds for integration 1000 to 5000?

Thank you for your help.

Solution

a)

The cdf of an exponential distribution is given by

F(x) = exp(-lambda*x)

As

F(1000) = 1 - 0.10 = 0.90

Then

exp(-lambda*1000) = 0.90

lambda = 0.000105361

Thus,

F(5000) = exp(-0.000105361*5000) = 0.59048857 [ANSWER]

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b)

For exponential distributions,

mean = 1/lambda = 1/0.000105361 = 9491.221581 [ANSWER]

standard deviation = mean = 9491.221581 [ANSWER]