Sum of random variable Could we think about joint distribution as sum of random variables?
Is the sum of random variables is the same thing as joint distribution of this variables?
Would be appreciated!
 A: Sum of Independent random variables is called convolution of probability distributions.
As others have pointed out, these two are entirely different things. To understand why, consider two independent standard normal distributions, $X$ and $Y$. Their joint distribution is a two dimensional gaussian with mean $(0,0)$ and covariance matrix $I_2$. But the sum of the two random variables will be a 1 dimensional gaussian with mean $E(X) + E(Y)$ and variance $Var(X)+Var(Y)$.
A: I think Mark L. Stone has a good answer. The answer is definitely NO.
I will give you another toy example to highlight the differences. Think about you have $2$ discrete binary random variables $X$ and $Y$, i.e., each of them can take $0$ or $1$ with different probability, for each random variable the probability mass function can be described with a $1 \times 2$ table.


*

*The sum is still a binary random variable but can take from $0$ to $2$, the probability mass function can be described it with a $1 \times 3$ table.

*The join distribution is not a binary random variable, but a $2 \times 2$ table.


EDIT:
To answer your question in the comment, I would extend the binary discrete example.
Say $X$ and $Y$ are iid, and 
$P(X=0)=0.2$, $P(X=1)=0.8$, 
$P(Y=0)=0.2$, $P(Y=1)=0.8$. 
Let $Z=X+Y$
Then 
$P(Z=0)=0.2*0.2=0.04$ ,
$P(Z=2)=0.8*0.8=0.64$,
$P(Z=1)=0.2*0.8+0.8*0.2=0.32$.
The mean of $Z$ is
$0*0.04+2*0.64+1*0.32=0.16$
