- #1

ChrisVer

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If I have two random variables [itex]X, Y[/itex] that are given from the following formula:

[itex] X= \mu_x \big(1 + G_1(0, \sigma_1) + G_2(0, \sigma_2) \big) [/itex]

[itex] Y= \mu_y \big(1 + G_3(0, \sigma_1) + G_2(0, \sigma_2) \big)[/itex]

Where [itex]G(\mu, \sigma)[/itex] are gaussians with mean [itex]\mu=0[/itex] here and std some number.

How can I find the covariance matrix of those two?

I guess the variance will be given by:

[itex]Var(X) = \mu_x^2 (\sigma_1^2+ \sigma_2^2)[/itex] and similarly for Y. But I don't know how I can work to find the covariance?

Could I define some other variable as :[itex]Z=X+Y[/itex] and find the covariance from [itex]Var(Z)= Var(X)+Var(Y) +2 Cov(X,Y)[/itex] ?

while [itex]Z[/itex] will be given by [itex]Z= (\mu_x+\mu_y) (1+ G_1 + G_2) [/itex]?

Then [itex]Var(Z)= (\mu_x+ \mu_y)^2 (\sigma_1^2+ \sigma_2^2)[/itex]

And [itex]Cov(X,Y) = \dfrac{(\mu_x+\mu_y)^2(\sigma_1^2+ \sigma_2^2)- \mu_x^2 (\sigma_1^2+ \sigma_2^2) - \mu_y^2(\sigma_1^2+ \sigma_2^2) }{2}=\mu_x \mu_y(\sigma_1^2+ \sigma_2^2) [/itex]

Is my logic correct? I'm not sure about the Z and whether it's given by that formula.

[itex] X= \mu_x \big(1 + G_1(0, \sigma_1) + G_2(0, \sigma_2) \big) [/itex]

[itex] Y= \mu_y \big(1 + G_3(0, \sigma_1) + G_2(0, \sigma_2) \big)[/itex]

Where [itex]G(\mu, \sigma)[/itex] are gaussians with mean [itex]\mu=0[/itex] here and std some number.

How can I find the covariance matrix of those two?

I guess the variance will be given by:

[itex]Var(X) = \mu_x^2 (\sigma_1^2+ \sigma_2^2)[/itex] and similarly for Y. But I don't know how I can work to find the covariance?

Could I define some other variable as :[itex]Z=X+Y[/itex] and find the covariance from [itex]Var(Z)= Var(X)+Var(Y) +2 Cov(X,Y)[/itex] ?

while [itex]Z[/itex] will be given by [itex]Z= (\mu_x+\mu_y) (1+ G_1 + G_2) [/itex]?

Then [itex]Var(Z)= (\mu_x+ \mu_y)^2 (\sigma_1^2+ \sigma_2^2)[/itex]

And [itex]Cov(X,Y) = \dfrac{(\mu_x+\mu_y)^2(\sigma_1^2+ \sigma_2^2)- \mu_x^2 (\sigma_1^2+ \sigma_2^2) - \mu_y^2(\sigma_1^2+ \sigma_2^2) }{2}=\mu_x \mu_y(\sigma_1^2+ \sigma_2^2) [/itex]

Is my logic correct? I'm not sure about the Z and whether it's given by that formula.

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