Probability - Gravner-73

Question

Assume that a light bulb lasts on average $100$ hours. Assuming exponential distribution, compute the probability that it lasts more than $200$ hours and the probability that it lasts less than $50$ hours.

Answer 1

Here $\text{mean} (\beta) = 100$

so $\lambda = \large \frac{1}{\beta} =  \large \frac{1}{100}$

$P(X \leq x) = 1 - \large  e^{- \lambda x}$

$P(X \leq 200) = 1 - e^{-2} $

$P(X \geq 200) = 1 - P(X \leq 200)  $

$= 1 - (1 - e^{-2}) = e^{-2}  \approx   0.1353 $

Probability the bulb lasts more than $200$ hours is $0.1353$

$P(X \leq 50) = 1 - e^{- \frac{1}{100} \times 50}  \approx 0.3935$

Probability the bulb lasts less than $50$ hours is $0.3935$