I was reading an article, and came across the following:
Purchase count follows a Poisson distribution with rate λ. In other words, the timing of these purchases is somewhat random, but the rate (in counts/unit time) is constant. In turn, this implies that the inter-purchase time at the customer level should follow an exponential distribution.
It's been quite a while since I did any statistics so I am struggling with the definitions of a Poisson distribution. What I understand by the "rate is constant" is that if a customer purchases 1 thing on average in a week, they purchase 4 things on average in a four-week period. Is this correct?
Where I believe I am confused is with the final sentence. Is this saying that the time between a customers purchases would grow exponentially as time goes on? Doesn't this contradict the idea that we have a constant rate of purchase?