Let the Nv by $($N$+1)$ input vector data matrix Dx be a finite size ensemble of a random process which generates random length$-($N$+1)$ input vectors x where $($xp$)$T denotes the pth row vector from the ensemble. This means that expected values are calculated as Nvpv p$=11$E$[$f$()]=$ f$()$ Nx x Similarly, let the Nv by M target vector data matrix Dt be a finite size ensemble of a random process which generates the corresponding random length$-$M target vectors t $.$ Here xp and tpdenote the pth waveforms or column vectors from the two ensembles. $($a$)$ If the $($N$+1)$ by $($N$+1)$ autocorrelation matrix R is defined as R $=$ E$[$xxT$],$ where E$[]$ denotes expected value, give r$($m$,$n$)$ in terms of xp$()$ and Nv $.$ Give R in terms of Nv and Dx $.($b$)$ If the $($N$+1)$ by M crosscorrelation matrix C is defined as C $=$ E$[$xtT$],$ give c$($m$,$i$)$ in terms of xp$(),$ tp$()$ and Nv $($Its a sum$).$ Give C in terms of Nv $,$ Dx $,$and Dt$.($c$)$ For the M$=1$ case, rewrite the linear network MSE, Nv N$+12$p pv p$=1$ n $11$E $=[$t $-$ w$($n$)$x $($n$)]$ N $=$ in terms of data matrices. Repeat for the M $>1$ case where Nv M N$+12$p pv p$=1$ i $1$ n $11$E $=[$t $($i$)-$ w$($i$,$ n$)$x $($n$)]$ N $==$ Hint, Nn$=1$tr$()=$ a$($n$,$n$)$