If x = y*y, and you know var(y), var(z), and cov(y,z), do I know cov(x,z)? If I know that x = y*y, and I know a whole of statistics pertaining to y, such as the variance and its covariance with other variables, can I analytically solve for the variance and covariance of x? Also, if I know a bunch of statistics regarding x, what can I know about y analytically?
Thanks for the help!
 A: Just complementing whuber comment, here's an example in R showing that changing the mean of y and z will change cov(y*y, z) even if var(y), var(z), and cov(y,z) remains the constant.
require(MASS)
covarianceMatrix<-matrix(c(3,2,2,4),2,2)
yz <- mvrnorm(100000, c(1,2), covarianceMatrix)
var(yz[, 1]) #variance y
var(yz[, 2]) #variance z
cov(yz[, 1], yz[, 2]) #covariance yz
cov(yz[, 1]*yz[, 1], yz[, 2]) #covariance yz

yz <- mvrnorm(100000, c(2,3), covarianceMatrix)
var(yz[, 1]) #variance y is approximatelly equal
var(yz[, 2]) #variance z is approximatelly equal
cov(yz[, 1], yz[, 2]) #covariance yz is approximatelly equal
cov(yz[, 1]*yz[, 1], yz[, 2]) #covariance yz is different

Output:
>     require(MASS)
>     covarianceMatrix<-matrix(c(3,2,2,4),2,2)
>     yz <- mvrnorm(100000, c(1,2), covarianceMatrix)
>     var(yz[, 1]) #variance y
[1] 2.999731
>     var(yz[, 2]) #variance z
[1] 3.987699
>     cov(yz[, 1], yz[, 2]) #covariance yz
[1] 1.994687
>     cov(yz[, 1]*yz[, 1], yz[, 2]) #covariance yz
[1] 4.018674
>     
>     yz <- mvrnorm(100000, c(2,3), covarianceMatrix)
>     var(yz[, 1]) #variance y is approximatelly equal
[1] 3.005912
>     var(yz[, 2]) #variance z is approximatelly equal
[1] 3.994272
>     cov(yz[, 1], yz[, 2]) #covariance yz is approximatelly equal
[1] 2.004748
>     cov(yz[, 1]*yz[, 1], yz[, 2]) #covariance yz is different
[1] 7.963221

A: I don't think so..
Cov(X,Z)=Cov(Y^2,Z)=E[Y^2*Z]-E[Y^2]E[Z]
Even if I assume you have E[Z] and E[Y], though you have not mentioned it, then E[Y^2] can be calculated using Var(Y) and E[Y]; but the first term E[Y^2*Z] can not be calculated using the given information.
