Question

Suppose that the time (in hours) required to repair a machine is an exponentially distributed random variable with parameter ????=0.5. What is
(a) the probability that a repair takes less than 4 hours?
(b) the conditional probability that a repair takes at least 12 hours, given that it takes more than 7 hours?

Answer 1

We need to define the variables,

So,

$F_x (x) = 1-e^{-\lambda x}\\F_x (x) = 1-e^{-0.5x}$

Therefore, the probability that the repair time is more than 4 horus can be calculate as,

$P(x>4)=1-P(x4)= 1-F_x(4)\\P(x>4) = 1-e^{-0.5*4}\\P(x>4) = 1-0.98\\P(x>4) = 0.018$

The probability that the repair time is more than 4 hours is 0.136

b) The probability that repair time is at least 12 hours given that the repair time is more than 7 hoirs is calculated as,

$P(x\geq 12|x>7)=P(X\geq7+5|x>7)\\P(x\geq12|x>7)=P(X\geq5)\\P(x\geq12|x>7)=1-P(x\leq 5)\\P(x\geq12|x>7)=1-e^{-0.5(2)}$

$P(x\geq 12|x>7)=0.6321$

The probability that repair time is at least 12 hours given that the repair time is more than 7 hours is 0.63