When we talk about $X$ being a RV with an exponential distribution $$f(x)= 1-e^{-\theta x} \,\,\,\,\, \text{for}\,\,\,\ \theta>0$$ we say that it describes the time between two events in a Poisson Process, where $\theta$ is the rate.
Does $X$ describe the time interval between two specific events? Or does it consider the time interval between any two events. So like if I have 3 events, does it consider the time interval between 1 and 2 and then between 2 and 3?
If the rate is constant, then does it mean that we actually know WHEN the event is going to happen? I don't understand how it can be random with a constant rate. If I have a machines that builds cars at a constant rate $\lambda$, then of course after a time $t$ I will have $t\cdot \lambda$.
Best Answer
In a Poisson process, the time between two subsequent events (or arrivals as some call them) has an exponential distribution. It's a random variable, so we can't know exactly when the next event will occur.
The rate parameter $\theta$ tells us how often on average the events come. It's the expected number of arrivals in one unit of time. It doesn't mean that the events happen exactly at a given rate, but in the long run, the average number of events that happened in one unit of time would be $\theta$.
Let's say we model the number of people waiting in line with a Poisson process with the rate $\theta=20$ and let our unit of time be one hour. That means that, on average, 20 people get in the line every hour (according to the model we're using).
The time elapsed from the moment one person got in line to the next person has an exponential distribution with the rate $\theta$. You can't predict when exactly the next person will get in line, but you can expect him to show up in about $3$ minutes ($\frac 1 {20}$ hours).