What is THE MAIN DIFFERENCE between the Binomial and the Poisson’s distributions?

Question

My professor just gave us the question as I wrote it above, and we seriously didn't get it , I mean what should we answer him with ? , did he ment the basic differences ? If he did , then why it mentioned "THE MAIN" he could have wrote only the "differences" ?? I hope I'll find the answer for this question. PS : if you wandering why we didn't ask the professor these questions directly? Will, my answer is : "We actually did asked him the same questions but he kept silence and never answered them "

Answer 1

They are completely different. If a Poisson random variable has mean $5$ and a binomial random variable has parameters $25, 1/3$, then the two models are not even close. However, there is a key similarity between poisson and binomial random variables that is when $\lambda=np$, with $n$ large and $p$ small, then the poisson and binomial distributions are can approximate one another. In this case, and in this case only, the main difference is that the poisson can take any non negative integer value (albeit with very small probability), while a binomial random variable can take only the integer values between $0$ and $n$.