I am trying to understand intuitively the connection between the 3. Here's what I think I understand so far
- I have read that the normal distribution can approximate the other two
- The Poisson distribution is the limiting case where the number of "trials" goes to infinity while the individual trial probability goes to zero much like how the formula for continuous compound interest is formed
- The normal distribution works as a good approximation when the number of trials is quite large
These 2 distributions, Poisson and binomial are clearly related, but I don't understand why taking a larger number of samples would affect the approximation using a normal distribution, or even why the binomial distribution is related to the normal distribution at all. Can someone help me with some intuition? (using limits would help for large number of trials n)
Without being too technical;
The binomial distribution is symmetrical when $p$, the probability of success, is 0.5, so when the binomial's $p$ does not stray too far away from 0.5, the binomial distribution can be modelled by the symmetrical Normal distribution.
The binomial distribution is skewed when $p$ is close to 0 or 1, so when the binomial's $p$ has such values. say below 0.1 or above 0.9, it can be modelled by the heavily skewed Poisson distribution.
The Poisson Distribution is a limiting case of the binomial distribution is proved here : http://www.oxfordmathcenter.com/drupal7/node/297
The normal is a valid approximation for the binomial for large n is proved here: http://www.real-statistics.com/binomial-and-related-distributions/relationship-binomial-and-normal-distributions/binomial-and-normal-distributions-advanced/
Further details are easily found on this website. For example : https://stats.stackexchange.com/questions/32405/how-is-poisson-distribution-different-to-normal-distribution