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

1) It models the lifetime of something thatdoes not agein 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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# I don't understand the exponential distribution at all

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