Hi people. Can you help me realize why this is true? I can tell you that $P\_i$ and $P\_j$ are probabilities, i.e. $0 \leq P\_i, P\_j \leq 1$.


$\displaystyle \sum\_{i=1}^\infty \sum\_{j=1}^\infty ijP\_iP\_j \leq \sum\_{i=1}^\infty \sum\_{j=1}^\infty j^2P\_jP\_i$.



Why does this inequality hold?
As Will Jagy said it is not true in general.
But assume $S=\sum\_{j=1}^\infty j^2 P\_j$ converges, and apparently you are assuming
$\sum\_{i=1}^\infty P\_i = 1$. Then the right side converges to $S$.
You also know that $i^2+j^2\ge 2ij$ (because $(i-j)^2\ge 0$).
Absolute convergence of the right side lets you rearrange it to
$\sum\_j \sum\_i j^2 P\_jP\_i = \sum\_i\sum\_j i^2P\_iP\_j$. So the right side
is $\frac{1}{2}\sum\_i\sum\_j (i^2+j^2)P\_iP\_j$, which is then greater than
or equal to the left.


