Calculus 4, please help!!!

Question

A lamp uses one bulb with an average lifespan of 1100 hours. We bought two bulbs, so once the first bulb burns out, we replace it with a second of the same type. Assuming we can model the probability of failure of these bulbs by an exponential density function with mean μ=1100, find the probability that we can keep the lamp lit longer than 1700 hours.

 

Thank you.

Answer 1

The probability of not lasting a time T (even using both bulbs) is

 

∫₀^T  p(t1)  ∫₀^T-t1 p(t2)  dt1  dt2

 

where  p(t) =  (1/u) exp(- t/u)   

 

Carrying the integration over  t2 first and then over t1  results in:

 

1 - exp(-T/u)  - (T/u) exp(-T/u)

 

Therefore, the probability of lasting  for time T  is   exp(-T/u) + (T/u) exp(-T/u)  

 

Plugging in u = 1100 and T = 1700   gives this probability as 0.543