Merge pull request #8 from HansiBansi/patch-7

Update Mitschrift_InfStat.tex

Mitschrift_InfStat.tex

@@ -4404,7 +4404,7 @@ \section{Likelihood-Ratio-Tests und das Neyman-Pearson Lemma}
 		\Prob(d=1\;|H_1) & = \Prob(d=1,d_{LR}=1\;|H_1) + \Prob(d=1,d_{LR}=0\;|H_1) \\
 		& = \Prob(d_{LR}=1\;|H_1)-\Prob(d=0,d_{LR}=1\;|H_1) + \Prob(d=1,d_{LR}=0\;|H_1) \\
 		& = \Prob(d_{LR}=1\;|H_1) - \int_{x\colon d(x)=0, d_{LR}(x)=1} f_1(x) dx + \int_{x\colon d(x)=1, d_{LR}(x)=0} f_1(x) dx \\
-		& \le \Prob(d_{LR}=1\;|H_1) - \frac{1}{c}\int_{x\colon d(x)=0, d_{LR}(x)=1} f_0(x) dx + \int_{x\colon d(x)=1, d_{LR}(x)=0} f_0(x) dx \\
+		& \le \Prob(d_{LR}=1\;|H_1) - \frac{1}{c}\int_{x\colon d(x)=0, d_{LR}(x)=1} f_0(x) dx + \frac{1}{c}\int_{x\colon d(x)=1, d_{LR}(x)=0} f_0(x) dx \\
 		& = \Prob(d_{LR}=1\;|H_1) - \frac{1}{c}\left(\Prob(d=0,d_{LR}=1\;|H_0)-\Prob(d=1,d_{LR}=0\;|H_0)\right) \\
 		& = \Prob(d_{LR}=1\;|H_1) - \frac{1}{c}\left(\Prob(d_{LR}=1\;|H_0)-\Prob(d=1,d_{LR}=1\;|H_0)-\Prob(d=1,d_{LR}=0\;|H_0)\right) \\
 		& = \Prob(d_{LR}=1\;|H_1) - \frac{1}{c}\left(\underbrace{\Prob(d_{LR}=1\;|H_0)}_{=\alpha}-\underbrace{\Prob(d=1\;|H_0)}_{=\alpha}\right) \\