Sum of n random variables with normal distribution

#1
I am struggling with this problem:

Independent random variables X_1, ... , X_n have normal distribution with mean 0 and variance 2. What is the distribution of random variable: X_1 + ... + X_n ?

I know that the sum od two random variables with normal distribution has normal distribution as well, but does it apply to infinite number of variables? Besides, when we add these up, it's going to be just one random variable so shouldn't it have single-point distribution?
 

Dason

Ambassador to the humans
#2
You don't have an infinite number of random variables - you have n random variables. If you know what the distribution of the sum of two normal random variables is then recognize that you can write this as

((((X1 + X2) + X3) + ... ) + Xn)

Since X1+X2 is the sum of two random normal variables you know the distribution of it. That sum itself gives you a single random variable - call it Z. Then (X1 + X2 + X3) = (X1 + X2) + X3 = Z + X3 which is the sum of two normal random variables. Use this logic to figure out the distribution of the sum of n normal random variables.
 
#3
Thank you.
So in my case it's the sum of (X_1 + ... + X_n-1) + X_n which may be written as Z + X and also have normal distribution.