Finding standard deviation given joint probabilities

Question

I'm trying to find the standard deviation of $Z = X + Y$ given the following table:

I'm getting $E[Z^2] = 36.31$ and $E[Z] = 5.45$, giving me a variance of

$Var[X] = 36.31 - (5.45)^2 = 6.6075$ and a standard deviation of $\sqrt{6.6075} = 2.57$.

I'm off by a bit on this question. Any help would be appreciated.

Answer 1

I think you need to check your calculations. For example, I get $E(Z) =6.15$, but I haven't checked $E(Z^2)$ yet. Would it be cheating to chuck this into an excel spreadsheet?

Answer 2

The expected value of $Z^2$ is $E(Z^2)=E((X+Y)^2)=E(X^2+2XY+Y^2)$.

$=E(X^2)+2E(XY)+E(Y^2)$

$cov(X,Y)=-0.2994\neq 0$, therefore

$E(XY)=E(X)E(Y)+cov(X,Y)=E(X)E(Y)-0.2994$

Some intermediate results:

$E(X^2)=3.75, \ E(Y^2)=23.44,\ E(X)=1.61,\ E(Y)=4.54,\ E(Z)=6.15,\ cov(X,Y)=-0.2994$