Rate of Poisson Process

Question

Can the rate of a poisson process greater than 1? Also, Can there be only one arrival maximum during T defined as interval in Poisson process?

Answer 1

Yes, the rate can be any number greater than zero. The number of arrivals in an interval is, again, a random variable that can take any nonnegative integer value. The number of arrivals will follow a Poisson distribution with a rate parameter, $\lambda T$, where $\lambda$ is the expected number of arrivals during one unit of time.

Answer 2

@jjet 's answer is correct. I'll add a little that might help your intuition.

Suppose the rate is 1 per minute. If you changed time units to hours the rate would be 60 per hour, but the process would still be the same. So the numerical value of the rate parameter depends on the time unit, so can have any positive value.

For your second question, a rate of 1 per hour doesn't mean that an hour can have just one occurrence. It means that the average rate is 1 per hour, but the arrivals can bunch up or spread out. That's just what the mathematics of Poisson process calculates for you.

Edit, in response to comment:

But, shouldn't we enforce that, there not be more than one arrival at max, in an interval so that we get to count them. So how this is enforced, if we allow multiple arrivals in an interval?

No. You can't "enforce" or "allow" anything.

Suppose you flip a fair coin over and over again. Then the "arrival rate" for heads is one per two tosses, but you will frequently see two heads in a row. A long run of tails (no arrivals) doesn't mean heads next is more likely - the coin has no memory of what happened last time.

When there's a storm that the newspapers and the insurance companies say is a once-in-a-decade occurrence, that doesn't mean there won't be another one next year. Nor does the fact that you've gone 10 years without one mean you're "due" for one soon.

Randomness is really hard to understand.