$5\mathrm{.}15.$ Duality between Gaussian broadcast and multiple access channels. Consider the following Gaussian BC and Gaussian MAC: Gaussian BC: Y$\_(1)=$g$\_(1)$x$+$Z$\_(1)$ and Y$\_(2)=$g$\_(2)$x$+$Z$\_(2),$ where Z$\_(1)$N$(0,1)$ and Z$\_(2)$ N$(0,1).$ Assume average power constraint P on x $.$ Gaussian MAC: Y$=$g$\_(1)$x$\_(1)+$g$\_(2)$x$\_(2)+$Z $,$ where Z$$N$(0,1).$ Assume the average sum$-$power constraint $\_($i$=1)^$n $($x$\_(1$i$)^(2)($m$\_(1))+$x$\_(2$i$)^(2)($m$\_(2)))<=$nP$,($m$\_(1),$m$\_(2))$in$[1$:$2^($nR$\_(1))]\backslash$times $[1$:$2^($nR$\_(2))].($a$)$ Characterize the $($private$-$message$)$ capacity regions of these two channels in terms of P$,$g$\_(1),$g$\_(2),$ and power allocation parameter $\backslash$alpha in$[0,1].($b$)$ Show that the two capacity regions are equal. $($c$)$ Show that every point $($R$\_(1),$R$\_(2))$ on the boundary of the capacity region of the above Gaussian MAC is achievable using random coding and successive can$-$ cellation decoding. That is$,$ time sharing is not needed in this case. $($d$)$ Argue that the sequence of codes that achieves the rate pairs $($R$\_(1),$R$\_(2))$ on the boundary of the Gaussian MAC capacity region can be used to achieve the same point on the capacity region of the above Gaussian BC$.$ Remark: This result is a special case of a general duality result between the Gauss$-$ ian vector BC and MAC presented in Chapter $9.$