Question

Z = time taken to perform an action Pdf of Z: fz (z) = e^(−y). If two actions occur at t =0, but the second action only begins after the first is finished. What is the probability that the second action takes less than t = 2?

Answer 1

Let be the time taken by the first and the second actions respectively. The second action can only start at after the first action is finished. The second action can complete only at

We want the probability that the second action completes by T=2, that is we want the probability that

the 2 actions are independent of each other. The pdfs of z1 and z2 are

Let be a transformation and be another transformation

The pdf of is given by

The probability that the second action takes less than t = 2 is