The time till failure of an electronic component has an Exponential distribution

Question

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.

Explanation / Answer

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]

********************

b)

For exponential distributions,

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

standard deviation = mean = 9491.221581 [ANSWER]

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