TITLE: Finding standard deviation given joint probabilities
QUESTION [0 upvotes]: 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.

REPLY [0 votes]: 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:
$E(X^2)=3.75, \ E(Y^2)=23.44,\ E(X)=1.61,\ E(Y)=4.54,\ E(Z)=6.15,\ cov(X,Y)=-0.2994$
{"set_name": "stack_exchange", "score": 0, "question_id": 1388712}