We need to define the variables,
So,
[tex]F_x (x) = 1-e^{-\lambda x}\\F_x (x) = 1-e^{-0.5x}[/tex]
Therefore, the probability that the repair time is more than 4 horus can be calculate as,
[tex]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[/tex]
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,
[tex]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)}[/tex]
[tex]P(x\geq 12|x>7)=0.6321[/tex]
The probability that repair time is at least 12 hours given that the repair time is more than 7 hours is 0.63
