(Discrete) Binomial Distribution

The number of successes in $n$ independent trials of a $\text{Bernoulli}(p)$ r.v., denoted as $X\sim \text{Binomial}(n, p)$

Its PDF is

$$p(x|n,p)=\binom{n}{x}p^x(1-p)$$

Approximation

Poisson Distribution

When $n$ is large and $np$ is of moderate size (perferably $np\le 7$), binomial distribution can be approximated by $\text{Poisson}(np)$

Hypergeometric Distribution

If $N\gt 10,000$ and $n/N\lt 1\%$, hypergeometric distribution can be approximated by binomial distribution.

Normal Distribution