Let $X$ be a random variable over $\mathbb{R}$ with density $P_X$. Assume a finite disjoint partition of $\mathbb{R}$, $Z_1,Z_2,\dots, Z_M$. That is, \begin{equation} \bigcup_i Z_i= \mathbb{R}\:\:\:\text{and} \:\:\: Z_i \cap Z_j =\phi \end{equation}
Let \begin{equation} Y=f(X) \end{equation} for some real values function $f$. Let $Q:\mathbb{R}\to \{v_1,v_2,\dots,v_M\}$ be a function with finite range space where, \begin{equation} \begin{aligned} Q&:\mathbb{R}\to \{v_1,v_2,\dots,v_M\}\\ &:y \to v_i \:\:\:\text{if}, y \in Z_i \end{aligned} \end{equation}
and define, \begin{equation} Z= Q(Y) \end{equation} Also define, \begin{equation} P_{kl}=P(Z=v_l|Q(X)=v_k) \end{equation} Then show that, \begin{equation} \mathbb{E}(XZ)=\displaystyle\sum\limits_{l=1}^Mv_l\displaystyle\sum\limits_{k=1}^MP_{kl}\displaystyle\int_{Z_k}xp(x)\,dx \end{equation} My try:
I am stuck as I don't know how to proceed. Since $X$ is continuous and $Z$ is discrete I know that the density won't exists, hence I am unable to proceed computing it.
Any help would be appreciated.
Thank you