From what I know about Poisson process, time interval is divided evenly and are independent of each other. The number of occurrences in a small time interval has an average of $\lambda\cdot h$ where $h$ is the length of that time interval.
Question: if in a time interval, there happens to be a lot more cases than normal, how I am supposed to count it with $\lambda$?
Here is the problem: Telephone calls arrive at an office according to a Poisson process on the average of two every 3 minutes. Let $X$ denote the waiting time (in minutes) until the first call that arrives after 10 a.m. I want to find the PDF of $X$.
The formula is given $f(x)=\frac{1}{\theta}e^\frac{-x}{\theta}$.
I am not sure how to calculate $\theta$.
Best Answer
Given that the call is on average of two every 3 minutes. the expected number of calls $\lambda$ would be $\frac{2}{3}$ per minute.
Given the formula $f(x)$=$\frac{1}{\theta}$$e^\frac{x}{\theta}$ and $\theta$=$\frac{1}{\lambda}$. You can see the relation.