Original practice · fully worked
Original variant: conditional survival after an elapsed operating time
The operating time T of a sensor follows an exponential distribution. Calibration shows that P(T exceeds 4 hours)=0.80. Given that the sensor is still operating at hour 4, determine the probability that it remains operating beyond hour 10.
- A
0.64000 - B
0.70000 - C
0.80000 - D
0.84000 - E
0.71554
Answer and calculation
Memorylessness turns the condition into six additional hours of survival. Scaling the four-hour survival gives 0.80⁽⁶⁄⁴⁾=0.71554, choice E.
Setup
Setup
Record that the exponential survival probability through four hours is 0.80.
e−4λ=0.80 Model
Model
By memorylessness, conditional survival from hour 4 to hour 10 is the unconditional probability of surviving six additional hours.
Pr(T>10∣T>4)=Pr(T>6)=e−6λ Compute
Compute
Scale the four-hour survival to six hours by raising 0.80 to the 6/4 power, giving 0.715542.
e−6λ=(e−4λ)6/4=0.801.5=0.715542 Answer
Answer
The conditional probability of operating beyond hour 10 is 0.71554, which is choice E.
0.71554(E) 