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(x<4)\\P(x>4)= 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