Question

consider the binomial distribution pn (n) for n = 16 and p = q = 1/2. what is the value of pn (n) for n = σn/2? what is the value of the product pn (n = n)σn?

Answer 1

$$P_N(n)=\frac{N!}{n!(N-n)!}p^nq^{N-n}\\p=q=\frac{1}{2}, \sigma_n=\sqrt{Npq}=\frac{\sqrt{N}}{2}\\P_N(n)=\frac{N!}{\frac{\sqrt{N}}{2}!(N-\frac{\sqrt{N}}{2})!}\frac{1}{2^N}$$

$$\\P_N(n)\sigma_n=\frac{N!}{\frac{\sqrt{N}}{2}!(N-\frac{\sqrt{N}}{2})!}\frac{1}{2^N}\frac{\sqrt{N}}{2}$$