I understand the poission approximation to the binomial distribution:

$\displaystyle e^{\lambda}{\lambda^k \over{k!}}$

where $\displaystyle \lambda=np=$mean number of success in a given population of size n

But I am confused how it is mapped to the Poission distribution:

$\displaystyle e^{\lambda t}{(\lambda t)^k \over{k!}}$

where $\displaystyle \lambda t=$mean number of success in a given population of size n

Now, how can $\displaystyle \lambda$ and $\displaystyle \lambda t$ mean the same thing???

2. Originally Posted by chopet
Now, how can $\displaystyle \lambda$ and $\displaystyle \lambda t$ mean the same thing???
Think of the Poisson variable $\displaystyle \lambda$ as a rate, so you need to multiply it by a given amount of time for a mean or expected amount of outcomes in that time period.

3. So we are treating the first case as t=1?

t=2 would mean we are dealing with 2 populations?

But the mean number per population should be independent of t right?
If in t=1 population, the mean is 30 successes.
Then in t=10 populations, the mean is still 30 successes.

Is my concept wrong?