I am given two random variables $Y_1$ and $Y_2$ and:


$E(Y_1) = 4$
$E(Y_2) = -1$
$V(Y_1) = 2$
$V(Y_2) = 8$


I am asked to find $Cov(Y_1,Y_1)$

I know $Cov(Y_1,Y_2) = E(Y_1Y_2)-E(Y_1)E(Y_2)$

I'm not sure if it is a typo, but it says $Cov(Y_1,Y_1)$ NOT $Cov(Y_1,Y_2)$

The answer is $2$, but i'm not sure how to get $E(Y_1Y_2)$ from the information given, or why it's asking for the Cov of the same variable?

Thanks for any help!

The short answer is "because they don't work," but that's kind of a copout.  This is actually quite a deep question.  What you're referring to is sphere packing in two dimensions, specifically the kissing number, and sphere packing is actually quite a sophisticated and active field of mathematical research (in arbitrary dimensions).  

Here's one answer, which isn't complete but which tells you why $6$ is a meaningful number in two dimensions.  The packing you refer to is a special type of packing called a lattice packing, which means it comes from an arrangement of regularly spaced points; in this case, the hexagonal lattice.  The number $6$ appears here because the hexagonal lattice has $6$-fold symmetry.  So a natural question might be whether one can find lattices in two dimensions with, say $7$-fold or $8$-fold symmetry, since these might correspond to circle packings with more circles around a given circle.  (Intuitively, we expect more symmetric lattices to give rise to denser packings and to packings where each circle has more neighbors.)

The answer is no: $6$-fold symmetry is the best you can do!  This is a consequence of the crystallographic restriction theorem.  The generalization of the theorem to $n$ dimensions says this: it is possible for a lattice to have $d$-fold symmetry only if $\phi(d) \le n$, where $\phi$ is Euler's totient function.

The generalization implies that you still cannot do better than $6$-fold symmetry in $3$ dimensions.  There are two natural lattice packings in $3$ dimensions, which both occur in molecules and crystals in nature and which both have $6$-fold symmetry, and it turns out that these are the densest sphere packings in $3$ dimensions.  It also turns out that they give the correct kissing number in $3$ dimensions, which is $12$ (see the wiki article).

In $4$ dimensions, the kissing number is $24$, and I believe the corresponding packing is a lattice packing coming from a lattice with $8$-fold symmetry, which is possible in $4$ dimensions.  In higher dimensions, only two other kissing numbers are known: $8$ dimensions, where the $E_8$ lattice gives kissing number $240$, and $24$ dimensions, where the Leech lattice gives kissing number $196560$!  These lattices are really mysterious objects and are related to a host of other mysterious objects in mathematics.

A great reference for this stuff, although it is a little dense, is Conway and Sloane's Sphere Packing, Lattices, and Groups (Wayback Machine).  Edit:  And for a very accessible and engaging introduction to symmetry in the plane and in general, I highly recommend Conway, Burgiel, and Goodman-Strauss's The Symmetries of Things.
