1. ## Binomial distribution

For the $B(n,p)$ distribution, by considering $\frac{p_{k}}{p_{k-1}}$, show that $p_{k}$ is largest when $k=[(n+1)p]$. This k is called the "mode" of the distribution.

2. Given

$
p_i = \frac {(n-i+1)!} {k!} p^i (1-p)^{n-i}
$

then

$
\displaystyle{\frac {p_k} {p_{k-1}} = \frac {\frac {(n-k+1)!} {k!} p^k (1-p)^{n-k}} {\frac {(n-(k-1)+1)!} {(k-1)!} p^{k-1} (1-p)^{n-(k-1)}}}
$

$
\displaystyle{=\frac {n-k+1} k \left( \frac {1-p} p \right)}
$

You want to find the point at which the value of $p_k$
goes from being greater than 1 to less than 1, because that is the point at which the probability of $p_{k} < p_{k-1}$

$
\displaystyle{\left( \frac {n-k+1} k \right) \left( \frac p {1-p} \right) = 1}
$

Solve for k.