- #1
noowutah
- 57
- 3
How do you solve the system of equations
[tex]e^{x_{1}+y_{1}}+e^{x_{1}+y_{2}}=a_{1}[/tex]
[tex]e^{x_{2}+y_{1}}+e^{x_{2}+y_{2}}=a_{2}[/tex]
[tex]e^{x_{1}+y_{1}}+e^{x_{2}+y_{1}}=b_{1}[/tex]
[tex]e^{x_{1}+y_{2}}+e^{x_{2}+y_{2}}=b_{2}[/tex]
x1, x2, y1, y2 are the variables for which I want to solve the equations, a1, a2, b1, b2 are known.
Context: I need to solve this in order to get the unknown maximum entropy joint probabilities
[tex]p_{ij}=e^{-1-x_{i}-y_{j}}[/tex]
[tex]\mbox{for the known marginal probabilities (}a_{i}\mbox{ and }b_{j}\mbox{).}[/tex]
i know there is way to do this in information theory, but I need to solve it algebraically.
[tex]e^{x_{1}+y_{1}}+e^{x_{1}+y_{2}}=a_{1}[/tex]
[tex]e^{x_{2}+y_{1}}+e^{x_{2}+y_{2}}=a_{2}[/tex]
[tex]e^{x_{1}+y_{1}}+e^{x_{2}+y_{1}}=b_{1}[/tex]
[tex]e^{x_{1}+y_{2}}+e^{x_{2}+y_{2}}=b_{2}[/tex]
x1, x2, y1, y2 are the variables for which I want to solve the equations, a1, a2, b1, b2 are known.
Context: I need to solve this in order to get the unknown maximum entropy joint probabilities
[tex]p_{ij}=e^{-1-x_{i}-y_{j}}[/tex]
[tex]\mbox{for the known marginal probabilities (}a_{i}\mbox{ and }b_{j}\mbox{).}[/tex]
i know there is way to do this in information theory, but I need to solve it algebraically.