Given: $A = X + Y + Z$, the parameter of $Y$ and $Z$ is $\mu$, the parameter of $X$ is $\lambda$. The coefficient of correlation of Y and Z is $\beta$, $X$ and $Y$ and $X$ and $Z$ are pairwise independent ; What is the mean and variance of A?
My attempt: let $W = Y+Z$
Var[W] = Var[Y] + Var[Z] + 2 Cov[Y,Z]
= var[Y] + var[Z] + 2( beta*(1/mu)*(1/mu) )
= (1/mu^2) + (1/mu^2) + 2beta/mu^2
var[A] = var[X] + var[W] + 2cov[X,W]
= var[X] + var[W] + 2( E[XW] + E[X]E[W] )
= var[X] + var[W] + 2( E[X]E[W] + E[X]E[W] ) since X is independent of Y and Z
=1/lambda^2 + (2+2beta)/mu^2 + 2((2+2beta)^2)/lambda^4
is this correct? I have a feeling its not...
oops forgot the mean,
should be E[A] = E[X] + E[Y] + E[Z] = 1/lambda + 2/mu