Question

Two drilling machines(machine A and machine B) can drill deep holes with different diameters on workpieces. The processing time of the two machines are random due to various workpiece materials. Suppose the time it takes machine A to complete drilling one hole is exponentially distributed with mean 1.5,and the time it takes machine B to complete drilling one hole is exponentially distributed with mean 1. If the two machines work independently: 1) what is the probability that machine A completes the work first? 2) What is the probability that machine B completes the work first? 3) Given that machine A completes first, what is the expected amount of time it takes machine A to complete the work? 4) Given that machine A completes last, what is the expected amount of time it takes machine A to complete the work?

Answer 1

machine A completes the work first means time taken by machine A is less than that of machine B, i.e. X<Y and similarly machine B completes the work first means X>Y

1.The probability that machine A completes the work first=1/(1+1.5)= 0.4

2.The probability that machine B completes the work first =1.5/(1+1.5)= 0.6

3.since exponential distribution has "Memoryless property" i.e. P(X>a+b|X>a)=P(X>a+b) ,the the expected amount of time it takes machine A to complete the work given that machine A completes first ,i.e. E(X|X<Y) is equal to E(X)=1.5

4.similarly given that machine A completes last, is the expected amount of time it takes machine A to complete the work,i.e. E(X|X>Y)=E(X)=1.5

calculation of cdf of Z when z>0