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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.
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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 $0.3935$
by Boss (36.7k points)
edited