The life of a bulb (in hours) is random variable with an exponential distribution $f (t) =\alpha e^{- \alpha t}, 0 \leq t \leq \infty.$ Find the probability that its value lies between 100 and 200 hours.

The life of a bulb (in hours) is random variable with an exponential distribution $f (t) =\alpha e^{- \alpha t}, 0 \leq t \leq \infty.$ The probability that its value lies between 100 and 200 hours is 

(A) $e^{-100\alpha} - e^{-200\alpha}$ 

(B) $e^{-100} - e^{-200}$ 

(C) $e^{-100\alpha} + e^{-200\alpha}$ 

(D) $e^{-200\alpha} - e^{-100\alpha}$ 

(GATE 2005) 

Answer: (A) $e^{-100\alpha} - e^{-200\alpha}$ 

Explanation: 

The required probability is 

$P(100 \leq X \leq 200)$  

$= \int\limits_{100}^{200}f(t)dt$  

$= \int\limits_{100}^{200}\alpha e^{-\alpha t}dt$ 

$= e^{-100\alpha} - e^{-200\alpha}$