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]=\frac{\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]=\frac{{\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`