What is a numerical example of $Var(X_1 + X_2) = Var(X_1) + Var(X_2)$ I need a numerical example to illustrate cases where $Cov(X_1, X_2) = 0$. Can you think of examples involving functions or matrices? 
 A: As a very simple example (maybe too simple?), consider $X,Y\in\{0,1\}$ with joint distribution defined by the table
  Y \ X   0    1    
  0     1/4  1/4 1/2
  1     1/4  1/4 1/2
        1/2  1/2   1

This table also displays the marginal distributions of $X$ and $Y$. First, check that $X$ and $Y$ are independent. For example,
$$
  \mathrm{Pr}(X=0,Y=0) = 1/4 = 1/2 \times 1/2 = \mathrm{Pr}(X=0)\,\mathrm{Pr}(Y=0) \, ,
$$
and so on. Now, compute the distribution of $Z=X+Y\in\{0,1,2\}$. For example,
$$
\mathrm{Pr}(Z=1) = \mathrm{Pr}(X=1,Y=0) + \mathrm{Pr}(X=0,Y=1) = 1/2 \, .
$$
Using these distributions, compute $\mathrm{Var}(X),\mathrm{Var}(Y)$, and $\mathrm{Var}(Z)$.
A: Obviously whenever $X_1,X_2$ are independent but I guess that's not the point.
My go-to for dependent rvs is $U$ uniform on $[0,1]$ and take $X_1 = \sin(2\pi U), X_2=\cos(2\pi U)$. This basically says that if you pick a point uniformly on the unit circle then the coordinate functions are uncorrelated. This fact boils down to showing 
$$
\int_0^{2\pi} \sin(t)\cos(t) \ dt = 0.
$$
A: A simple example of an uncorrelated but dependent pair:
$X_1=(0,0.1,\dots,1)\cdot \pi$ 
and
$X_2=\sin(X_1)$
Edit: Since it is not particularly easy to work with trigonometric functions, you might as well work with a triangle:
$X_1=(-2,-1,0,1,2)$ with mean 0 and
$X_2=2-|X_1|$
A: This will be (approximately) true of any two independent variables. If you're ok with cov(x,y) being nearly but not actually 0, generating an example should be trivial:
set.seed(123)
N=1000
x = rnorm(N)
y = rnorm(N)
cov(x,y)

0.0865909

As N approaches infinity, your covariance will approach zero.
A: You need independent RV's since then the covariance is identically equal to zero. An example might be the following distributions 
$$ \begin{align} f_Y (y)= 2y\ \text{for}\ 0<y<1 \\ f_X(x)= 2x\ \text{for}\ 0<x<1 \\ f_(x,y)=4xy \end{align} $$
Try it!
