# Are the sums of independent RVs necessarily independent of each other?

• August 28th 2013, 08:07 AM
Are the sums of independent RVs necessarily independent of each other?
Suppose that $X_1,X_2, \ldots , X_n$ are i.d.d. to some distributions $( \mu _X , \sigma ^2 _X )$ and $Y_1, \ldots , Y_m$ are i.d.d. to $( \mu _Y , \sigma _Y ^2 )$.

If all the X's and Y's are indepedent to one aother, does that also mean that the sums $X_1 + \ldots + X_n$ is also independent to $Y_1 + \ldots + Y_m$?

So I'm trying to prove it by doing it striaght from the definition:

$Pr[ X_1 + \dots + X_n \cap Y_1 + \dots + Y_m ]$ and try to get it to become $Pr[X_1 + \dots + X_n ] \cdot Pr[Y_1 + \dots + Y_m ]$

But I'm getting stuck on the left hand side, not sure how to break it down...