91Ó°ÊÓ

$S=$Signal Sent

$R=$Signal received

$E\left[R\right]=?$

From the information, observe that the signal is sent from point $A$According to the distribution $S~N\left(\mathrm{μ},{\mathrm{σ}}^2\right)$

While it gets to the point $B$, the little error $\mathrm{Î\mu }$has been made to the original signal.

So, the received signal $S$can be written as $R=S+\mathrm{Î\mu }$

Where, $\mathrm{Î\mu }~N(0,1)$

Signal that is sent is independent from the error.

Calculate $E\left(R\right),$

$E\left(R\right)=E\left(S\right)+E\left(\mathrm{Î\mu }\right)$

$=\mathrm{μ}+0$

$=\mathrm{μ}$

Hence, the value of$E\left[R\right]$is$\mathrm{μ}.$

$S=$Signal sent

$R=$Signal received

$Var\left(R\right)=?$

Calculate variance:

$V\left(R\right)=V\left(S\right)+V\left(\mathrm{Î\mu }\right)$

role="math" localid="1647434029229" $={\mathrm{σ}}^2+1$

Hence, the value of$Var\left(R\right)$is${\mathrm{σ}}^2+1$

$S=$Signal Sent

$R=$Signal received

Yes.

$R$is normally distributed.

Since $R$can be written was the sum of two independent normally distributed random variables.

Hence, the sum is also normally distributed random variables with parameters are as follows:

$N~N\left(\mathrm{μ},{\mathrm{σ}}^2+1\right)$

Therefore,$R$is normally distributed.

$S=$Signal sent

$R=$Signal received

$Cov(R,S)=?$

Calculate

$\begin{array}{r}=\mathrm{Var}\left(S\right)+\mathrm{Cov}(\mathrm{Î\mu },S)\end{array}$

role="math" localid="1647434711595" $=\mathrm{Var}\left(S\right)$

$={\mathrm{σ}}^2$

Therefore,$Cov(R,S)$is ${\mathrm{σ}}^2.$