Correct Poisson Distribution Formulation I have a Poisson formulation with the following parameters:


*

*I see 90 people in a city an hour

*1.2% of people in the city are Indian


How would I calculate:


*

*The maximum amount of time before I would see any Indian person with the highest probability

*The amount of Indian people I would see in an hour with the highest probability 

 A: Let $X$ denote the  the time till the first occurrence of the event of meeting an Indian person. As I understand, you are looking for the value of $t$ that would maximize the probability $P(X=t)$. Since time is a continuous variable, we can only look at $P(t < X <t+dt)$ - the probability of being close to $t$ (because $P(X=t)=0$ for continuous variables), or equivalently the mode of the random variable $X$.
You are assuming a Poisson process with rate $\lambda=1/(90*0.012)$, so the distribution of 
$X \sim Exp(\lambda)$.
The density function is $f(t)=\lambda \exp(-\lambda t)$, which is monotone decreasing in $t$, so the mode is $0$. 
So the hour with the highest probability of seeing the first Indian person is the first hour (and the instant with the highest probability is when you start the observations). However the expected number of Indian people you see in an hour does not depend on the time of observation - this is a basic assumption of the Poisson process. It will always be $\lambda$ events in one unit of time.
