# Finding the mean and variance of the sum of three exponentially distributed random variables

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

= (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

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$\text{cov}(XY)=\Bbb E(XY)\color{red}-\Bbb E(X)\Bbb E(Y)$. –  David Mitra Feb 21 '13 at 13:00
And the mean of A? –  Did Feb 21 '13 at 13:03
yes in the variance of W, i used the definition of coefficient of correltion which states that cov = roe(std(Y)*std(Z)) –  user63303 Feb 21 '13 at 13:03
Hint: $$\text{var}(X+Y+Z)=\text{var}(X)+\text{var}(Y)+\text{var}(Z)+2\text{cov}(X,Y)+2‌​\text{cov}(X,Z)+2\text{cov}(Y,Z).$$ Of the $6$ terms on the right side, how many have values that you know already or can figure out from the given information? –  Dilip Sarwate Feb 21 '13 at 14:04