Odds Ratios and Mode 10/30

The odds of k successes relative to (k-1) successes are:

$$\frac{P(X=k)}{P(X=k-1)}=\frac{n-k+1}{k}\frac{p}{q}$$

This is very useful for computing by recursion the probability mass of the binomial.

Property:
For X a B(n,p) random variable with probability of success p neither 0 or 1, then as k varies from 0 to n, P(X=k) first increases monotonically and then decreases monotonically, (it is unimodal)  reaching its highest value when k is the largest integer less or equal to (n+1)p (=floor(n+1)p).

Proof:

$$P(X=k) \geq P(X=k-1)$$

is equivalent to

$$(n-k+1)p\geq k(1-p)\; iff\; (n+1)p \geq k$$

The value where the the probability mass function takes on its maximum is called the mode.