Question

g SupposeXis a Gaussian random variable with mean 0 and varianceσ2X. SupposeN1is a Gaussian random variable with mean 0 and varianceσ21. SupposeN2is a Gaussianrandom variable with mean 0 and varianceσ22. AssumeX,N1,N2are all independentof each other. LetR1=X+N1R2=X+N2.(a) Find the mean ofR1andR2. That is findE[R1] andE[R2].(b) Find the correlationE[R1R2] betweenR1andR2.(c) Find the variance ofR1+R2.

Answer 1

a. $X$, $N_1$, and $N_2$ each have mean 0, and by linearity of expectation we have

$E[R_1]=E[X+N_1]=E[X]+E[N_1]=0$

$E[R_2]=E[X+N_2]=E[X]+E[N_2]=0$

b. By definition of correlation, we have

$\mathrm{Corr}[R_1,R_2]=\dfrac{\mathrm{Cov}[R_1,R_2]}{{\sigma_{R_1}}{\sigma_{R_2}}}$

where $\mathrm{Cov}$ denotes the covariance,

$\mathrm{Cov}[R_1,R_2]=E[(R_1-E[R_1])(R_2-E[R_2])]$

$=E[R_1R_2]-E[R_1]E[R_2]$

$=E[R_1R_2]$

$=E[(X+N_1)(X+N_2)]$

$=E[X^2]+E[N_1X]+E[XN_2]+E[N_1N_2]$

Because $X,N_1,N_2$ are mutually independent, the expectation of their products distributes over the factors:

$\mathrm{Cov}[R_1,R_2]=E[X^2]+E[N_1]E[X]+E[X]E[N_2]+E[N_1]E[N_2]$

$=E[X^2]$

and recall that variance is given by

$\mathrm{Var}[X]=E[(X-E[X])^2]$

$=E[X^2]-E[X]^2$

so that in this case, the second moment $E[X^2]$ is exactly the variance of $X$,

$\mathrm{Cov}[R_1,R_2]=E[X^2]={\sigma_X}^2$

We also have

${\sigma_{R_1}}^2=\mathrm{Var}[R_1]=\mathrm{Var}[X+N_1]=\mathrm{Var}[X]+\mathrm{Var}[N_1]={\sigma_X}^2+{\sigma_{N_1}}^2$

and similarly,

${\sigma_{R_2}}^2={\sigma_X}^2+{\sigma_{N_2}}^2$

So, the correlation is

$\mathrm{Corr}[R_1,R_2]=\dfrac{{\sigma_X}^2}{\sqrt{\left({\sigma_X}^2+{\sigma_{N_1}}^2\right)\left({\sigma_X}^2+{\sigma_{N_2}}^2\right)}}$

c. The variance of $R_1+R_2$ is

${\sigma_{R_1+R_2}}^2=\mathrm{Var}[R_1+R_2]$

$=\mathrm{Var}[2X+N_1+N_2]$

$=4\mathrm{Var}[X]+\mathrm{Var}[N_1]+\mathrm{Var}[N_2]$

$=4{\sigma_X}^2+{\sigma_{N_1}}^2+{\sigma_{N_2}}^2$