Let $X$ and $\theta$ be two random variables, and $x$ and $\theta$ are two number. We know (a) if both of $X$ and $\theta$ are discrete random variables, then $$ P_{\theta \mid X}\left(\theta_{0} \mid x ight)=\frac{P_{X \mid \theta}\left( \mid \theta_{0} ight) P_{\theta}\left(\theta_{0} ight)}{\sum_{\theta_{1}} P_{X \mid \theta]\left( \mid \theta_{1} ight) P_{\theta}\left(\theta_{1} ight)} $$ (b) $X$ is continuous and s\theta$ is discrete random variable, then $$ P_{\theta \mid X}\left(\theta_{0} \mid x ight)=\frac{f_{X \mid \theta}\left( \mid \theta_{0} ight) P_{\theta}\left(\theta_{0} ight)}{\sum_{\theta_{1}} f_{X \mid \theta}\left( \mid \theta_{1} ight) P_{\theta}\left(\theta_{1} ight)} $$ (c) if both of $X$ and $\theta$ are continuous random variables, then $$ f_{\theta \mid X}\left(\theta_[0] \mid x ight)=\frac{f_{X \mid \theta}\left( \mid \theta_{0} ight) f_{\theta}\left(\theta_{0} ight)}{\int_{\theta_{1}} f_{X \mid \theta}\left( \mid \theta_{1} ight) f_{\theta}\left(\theta_{1} ight) d \theta_{1}} $$ EQuestion: please write the formula for the case that $X$ is discrete and $\thetas is continuous. And roughly explain the notations appearing in your answer. SP.PB. 111