# I don't understand the exponential distribution at all

This is driving me completely crazy!!

QUESTION 1: There are two interpretations I find for the exponential distribution:

1) It models the lifetime of something that does not age in the sense that the probability of functioning yet another time unit does not depend on its current age. So, P(X > x+y | X > y) = P(X > x).

2) It arises naturally when modeling the time between independent events that happen at a constant average rate (whatever that means). For instance the rate of incoming phone calls.

How do these two imply each other?? How are they equivalent? I.e., if it models something that doesn't age, then how does it also model the time between events with "constant average rate"? And if it models the time between events with "constant average rate," then how does it model something that doesn't age?

QUESTION 2: For X~exp(lambda), how is the lambda a "rate"?

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Question 2 first:
Lambda = average number of events per unit of time
Such as the average number of accidents per month

Question 1:
"Does not age" means that just because something has not happened in a while it is not any more likely to happen.
"Constant average rate" means Lambda is not changing.

The exponential distribution gives the probability for the waiting time between Poisson events. (From Stewart's Probability for Risk Management)