begin align a_ 00 a_ 00 a_ 10 a_ 20 a_ 30 a_ 00 a_ 01 a_ 02 a_ 03 sum_

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\begin{align}
f(0,0) = p(0,0) = a_{00} \\
f(1,0) = p(1,0) = a_{00} + a_{10} + a_{20} + a_{30} \\
f(0,1) = p(0,1) = a_{00} + a_{01} + a_{02} + a_{03} \\
f(1,1) = p(1,1) = \sum_{i=0}^{3} \sum_{j=0}^{3} a_{ij} \\
\end{align}
\begin{align}
f_x(0,0) = p_x(0,0) = a_{10} \\
f_x(1,0) = p_x(1,0) = a_{10} + 2a_{20} + 3a_{30} \\
f_x(0,1) = p_x(0,1) = a_{10} + a_{11} + a_{12} + a_{13} \\
f_x(1,1) = p_x(1,1) = \sum_{i=1}^{3} \sum_{j=0}^{3} a_{ij} i \\
f_y(0,0) = p_y(0,0) = a_{01} \\
f_y(1,0) = p_y(1,0) = a_{01} + a_{11} + a_{21} + a_{31} \\
f_y(0,1) = p_y(0,1) = a_{01} + 2a_{02} + 3a_{03} \\
f_y(1,1) = p_y(1,1) = \sum_{i=0}^{3} \sum_{j=1}^{3} a_{ij} j \\
\end{align}
\begin{align}
f_{xy}(0,0) = p_{xy}(0,0) = a_{11} \\
f_{xy}(1,0) = p_{xy}(1,0) = a_{11} + 2a_{21} + 3a_{31} \\
f_{xy}(0,1) = p_{xy}(0,1) = a_{11} + 2a_{12} + 3a_{13} \\
f_{xy}(1,1) = p_{xy}(1,1) = \sum_{i=1}^{3} \sum_{j=1}^{3} a_{ij} i j \\
\end{align}