how to differentiate to get the maximum after using the binomial RV
this is an example from the wiki
Now suppose we had only one coin but its p could have been any value 0 ≤ p ≤ 1. We must maximize the likelihood function:
- $ L(\theta) = f_D(\mathrm{H} = 49 \mid p) = \binom{80}{49} p^{49}(1-p)^{31} $
over all possible values 0 ≤ p ≤ 1.
One way to maximize this function is by differentiating with respect to p and setting to zero:
- $ \begin{align} {0}&{} = \frac{\partial}{\partial p} \left( \binom{80}{49} p^{49}(1-p)^{31} \right) \\ & {}\propto 49p^{48}(1-p)^{31} - 31p^{49}(1-p)^{30} \\ & {}= p^{48}(1-p)^{30}\left[ 49(1-p) - 31p \right] \\ & {}= p^{48}(1-p)^{30}\left[ 49 - 80p \right] \end{align} $
