Question

Suppose that the time (in hours) required to repair a machine is an exponentially distributed random variable with parameter ????=0.6. What is (a) the probability that a repair takes less than 8 hours? (b) the conditional probability that a repair takes at least 7 hours, given that it takes more than 4 hours?

Answer 1

Answer:

a) 0.9917

b) 0.1652          

Step-by-step explanation:

We are given the following in the question:

The time for repair follows an exponential distribution.

$\lambda = 0.6$

a) P(repair takes less than 8 hours)

$P(x\leq 8)\\=\displaystyle\int^8_0(0.6)e^{-0.6x}dx\\\\=\big[e^{-0.6x}\big]^8_0\\\\=-(e^{-0.6(8)}-e^{-0.6(0)})\\=0.9917$

0.9917 is the probability that a repair takes less than 8 hours.

b) the conditional probability that a repair takes at least 7 hours, given that it takes more than 4 hours

$P(x\geq 7|x\geq 4) = \dfrac{P(x\geq 7)}{P(x\geq 4)}\\\\=\dfrac{1-\int^{7}_00.6e^{-0.6x}dx}{1-\int^{4}_00.6e^{-0.6x}dx}\\\\=\dfrac{1-(-e^{-0.6(7)}+e^{0})}{1-(-e^{-0.6(4)}+e^{0})}\\\\=0.1652$

thus, 0.1652 is the conditional probability that a repair takes at least 7 hours, given that it takes more than 4 hours.