2
$\begingroup$

If X is the number of years a computer works, and it follows an exponential distribution with a lambda 3, what's the probability a computer will work in 8 years?

I'm not sure that I'm going about this the right way and would love to get feedback. I'm following this formula: $P(X \leq x) = 1 - e^{-\lambda x}$ When plugging in values for x=8 and lambda=3, I get that $P(X\leq 8 ) = .9999$...; and to get P(X>8) I subtracted this from 1. Final answer .000000000378

Is this correct? It seems like a very small probability and not sure that I'm going about it the right way. Any ideas are greatly appreciated. Thank you!

$\endgroup$
1
$\begingroup$

Seems that the approach is right. I haven't checked your final computations. Keep in mind that since the rate $\lambda$=3 the mean lifetime is 1/3 years. So lasting 8 years is a very long time and therefore P(X>8) should be very small. Note that 8 years is 24 average lifetimes.

$\endgroup$
  • 2
    $\begingroup$ A mean lifetime of 1/3 years is so unrealistic for most computers that we ought to use common sense (as employed admirably by the OP) and interpret "lambda 3" as being a mean of 3 years. $\endgroup$ – whuber Jan 2 '18 at 1:37
  • $\begingroup$ We don't know what the OP thought. It was the rate parameter as stated on the question. I agree that very few computers breakdown in 1/3 of year. $\endgroup$ – Michael Chernick Jan 2 '18 at 1:40
  • 2
    $\begingroup$ Michael, the word "rate" appears nowhere in the question. I agree that the OP was using "lambda" as if it were a rate--but isn't that error precisely the issue that needs to be addressed, rather than being compounded by repeating it? $\endgroup$ – whuber Jan 2 '18 at 13:20
  • 1
    $\begingroup$ I don't understand what you're referring to. "Too small" compared to what? Regardless, between the two competing interpretations of $\lambda$ only one is plausible. $\endgroup$ – whuber Jan 2 '18 at 17:19
  • 1
    $\begingroup$ "Low" implies a comparison: it has no absolute meaning. The value, by the way, is close to 7%. $\endgroup$ – whuber Jan 2 '18 at 20:37

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.