Question

The life of a bulb (in hours) is a random variable with an exponential distribution $f\left(t\right)=\alpha e^{-\alpha t},0\le t\le \infty .$ 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. $e^{-200\alpha }-e^{-100\alpha }$

Answer 1

The correct option is A $e^{-100\alpha }-e^{-200\alpha }$

Let t : life of bulb

$P(100\le t\le 200)={\int }_{100}^{200}f\left(t\right)dt={\int }_{100}^{200}\alpha e^{-\alpha t}.dt$

$={(-e^{-\alpha t})}_{100}^{200}=(e^{-100\alpha }-e^{-200\alpha })$