Bernoulli Distribution Example - Likelihood Ratio

Example) I have no idea where I found this example, sorry!

$Y_{1},...,Y_{n}$ denote a random sample from Bernoulli $P(Y_{i}|p)=p^{y_{i}}(1-p)^{1-y_{i}}$ , where $y_{i}$=0 or 1. Suppose $H_{0}:P=P_{0}$ , $H_{1}:P=P_{a}$, where $P_{0} <$ $P_{a}$

(a) Show that  $\frac{L(P_{o})}{L(P_{a})}=[\frac{P_{0}\cdot (1-P_{a})}{(1-P_{0}) \cdot P_{a}}]^{\sum y_{i}}\cdot (\frac{1-P_{0}}{1-P_{a}})^n$ 

(b) Argue that $\frac{L(P_{o})}{L(P_{a})}$ < K iff $\sum y_{i} < k$ 

$\triangleright$ Solution (a)

$\frac{L(P_{o})}{L(P_{a})}=\frac{P_{0}^{\sum y_{i}}\cdot (1-P_{0}^{\sum y_{i}})}{P_{a}^{\sum y_{i}}\cdot (1-P_{a})^{n-\sum y_{i}}}$ $=\frac{\frac {P_{0}}{1-P_{0}}^{\sum y_{i}}\cdot (1-P_{0})^n}{\frac {P_{a}}{1-P_{a}}^{\sum y_{i}}\cdot (1-P_{a})^n}$ $= [\frac {P_{0}(1-P_{a})}{P_{a}(1-P_{0})}]^{\sum y_{i}}\cdot (\frac{1-P_{0}}{1-P{a}})$

$\triangleright$ Solution (b)

$\frac{L(P_{o})}{L(P_{a})} \leq k \Leftrightarrow \log \frac{L(P_{o})}{L(P_{a})}\leq \log k$ $\Leftrightarrow \sum y_{i} \log A +n\cdot \log B \leq \log k$ 

where A=$\frac {P_{0}(1-P_{a})}{P_{a}(1-P_{0})}$, B=$\frac{1-P_{0}}{1-P{a}}$ from (a), $\Rightarrow \sum y_{i} \log \frac {p_{0}(1-P_{a})}{P_{a}(1-P_{0})}\leq \log k - n\cdot \log \frac {1-P_{a}}{1-P_{0}}$
Since $P_{0} <$ $P_{a}$ $\Rightarrow \frac {P_{0}}{P_{a}} < 1$ $\Rightarrow \log \frac {p_{0}(1-P_{a})}{P_{a}(1-P_{0})} \leq \log 1 = 0$ 

$\therefore \sum y_{i}$ $\large \geq \frac{\log k - n\cdot \log \frac {p_{0}(1-P_{a})}{P_{a}(1-P_{0})}}{\log (\frac {p_{0}(1-P_{a})}{P_{a}(1-P_{0})})}$ = (set) 0