Statistics: Two random variables equal in distribution?

[kingwinner]

Homework Statement

Let X1,X2,X3,Y1,Y2,Y3 be random variables.
If X1 and Y1 have the same distribution,
X2 and Y2 have the same distribution,
X3 and Y3 have the same distribution,
then is it true that X1+X2+X3 and Y1+Y2+Y3 will have the same distribution? Why or why not?


2. Homework Equations /concepts
Probability & Statistics

The Attempt at a Solution

Intuition suggests that it does, but I can't think of a way of proving it...

Any help is appreciated!

 

[Mark44]

I don't know the answer, but here are some thoughts. A simpler problem that gets to the heart of this matter has X1, X2, Y1, and Y2 as random variables, with X1 and Y1 having the same distribution, and X2 and Y2 having the same distribution.

What is the distribution of X1 + X2? What is the distribution of Y1 + Y2? If both of these exist, are they equal?

Does it make any sense to add together two r.v.'s with different distributions? E.g., if X1 ~N(0, 1) and X2 ~ N(100, 5), can we say anything about the distribution of X1 + X2? For example, if X1 represents IQ test scores, and X2 represents weights of people in some group, does X1 + X2 have any meaning at all? It doesn't seem to me that it does.