In the case of N large (>15) and p=0.8, which is the best approximation for a discrete binomial distribution? I read that a normal distribution approximates a binomial distribution only if N is large but p is around 0.5 (for example 0.6). Could I approximate a binomial distribution to an exponential one?
Thank you.
The normal approximation does not depend on $p\sim 0.5$. From the central limit theorem, in the limit of fixed $p$ and large $N\rightarrow\infty$, the binomial distribution approaches a Gaussian distribution, with the parameters given by $\mu=Np$, $\,\sigma^2=Np(1-p)$. Use the formula
$$P_n\approx\int_{n-0.5}^{n+0.5}\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}}dx$$
to get probabilities for discrete values of $n$ between $1$ and $N-1$. For $n=0$ and $N$, replace the lower bound or upper bound by $\pm\infty$. This ensures that the probabilities $P_n$ are normalized, and gives accurate values if both $Np\gg 1$ and $N(1-p)\gg 1$. In case only one side is $\gg 1$ while the other side is not, use the Poisson approximation. If neither side is $\gg 1$, use the exact binomial distribution instead. You can always compare the approximate probabilities with the exact ones to see if the approximation is satisfactory.
PS: for small $N$, the parameters $\,\mu\,$ and $\,\sigma\,$ can be adjusted to achieve a better fit using the formulas $\,\mu=Np\,$ and $\,\sigma^2=Np(1-p)\,$ as a good initial guess.