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.

a. [tex]X[/tex], [tex]N_1[/tex], and [tex]N_2[/tex] each have mean 0, and by linearity of expectation we have[tex]E[R_1]=E[X+N_1]=E[X]+E[N_1]=0[/tex][tex]E[R_2]=E[X+N_2]=E[X]+E[N_2]=0[/tex]b. By definition of correlation, we have[tex]\mathrm{Corr}[R_1,R_2]=\dfrac{\mathrm{Cov}[R_1,R_2]}{{\sigma_{R_1}}{\sigma_{R_2}}}[/tex]where [tex]\mathrm{Cov}[/tex] denotes the covariance,[tex]\mathrm{Cov}[R_1,R_2]=E[(R_1-E[R_1])(R_2-E[R_2])][/tex][tex]=E[R_1R_2]-E[R_1]E[R_2][/tex][tex]=E[R_1R_2][/tex][tex]=E[(X+N_1)(X+N_2)][/tex][tex]=E[X^2]+E[N_1X]+E[XN_2]+E[N_1N_2][/tex]Because [tex]X,N_1,N_2[/tex] are mutually independent, the expectation of their products distributes over the factors:[tex]\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][/tex][tex]=E[X^2][/tex]and recall that variance is given by[tex]\mathrm{Var}[X]=E[(X-E[X])^2][/tex][tex]=E[X^2]-E[X]^2[/tex]so that in this case, the second moment [tex]E[X^2][/tex] is exactly the variance of [tex]X[/tex],[tex]\mathrm{Cov}[R_1,R_2]=E[X^2]={\sigma_X}^2[/tex]We also have[tex]{\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[/tex]and similarly,[tex]{\sigma_{R_2}}^2={\sigma_X}^2+{\sigma_{N_2}}^2[/tex]So, the correlation is[tex]\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)}}[/tex]c. The variance of [tex]R_1+R_2[/tex] is[tex]{\sigma_{R_1+R_2}}^2=\mathrm{Var}[R_1+R_2][/tex][tex]=\mathrm{Var}[2X+N_1+N_2][/tex][tex]=4\mathrm{Var}[X]+\mathrm{Var}[N_1]+\mathrm{Var}[N_2][/tex][tex]=4{\sigma_X}^2+{\sigma_{N_1}}^2+{\sigma_{N_2}}^2[/tex]