Here's a question we got for homework:
It is given that at a certain bank there's 50-50 chance that when you enter there's: - no one waiting in line - there's one man waiting, in which case the waiting time is exponentially distributed. What is the CDF of the total waiting tine? Instruction: Let X be the total waiting time, Y the number of people waiting. For x>=0, use the total probability theorem for the CDF of X Notice that X is not discrete nor continuous, but a mix of both.
Here's what I thought. As specified, Y can be either 0 or 1 people waiting. If there's no one waiting the waiting time is 0 which means P(X<=x) = 1 for all x>=0. If there's one man waiting then P(X<=x) = 1 − e^(−λx).
So, by the law of total probability,
P(X<=x) = P(X<=x|Y=1)P(Y=1) + P(X<=x|Y=0)P(Y=0)
Am I right so far? If I am right, then what is P(X<=x|Y=1)? Are the two variables independent?