sum of two random variable if X is a Binomial random variable (n,p), then :
X=Y1+Y2+Y3+Y4......Yn, where Y1,Y2....Yn are Bernoulli random variable

So, What exactly is this sum of multiply random variable means?
I know that Binomial random variable is just to repeat Bernoulli random variable n times, so the summation of two random variable means they are independent, they share the same sample space or what?


Update:
So, Z=X+Y, does this mean that Z=3  implies X=1 +Y=2 or X=2+Y=1 or X=0+Y=2  ?
 A: The sum of a random variable means that we create a new random variable $$P = Q + R$$ such that when you draw a value from $P$ it is equal to the sum of drawing a value from $Q$ and drawing a value from $R$.
Note:


*

*The random variables $P$ and $Q$ can be dependent or independent.

*They do not need to share the same sample space. E.g. let $P$ = the total number of items bought on a shopping trip, $Q$ = the number of eggs bought, $R$ = the number of chickens bought, and so on. Each of the variables that's being summed can have its own range and distribution.

*You can do algebra like $P = Q + R \implies P - R = Q$, and in the case where variables are discrete if you know $P$ then this constrains $Q$ and $R$. So in your case if $Z=3$ then you can have $(X,Y)=(0,3)$ or $(1,2)$ or $(2,1)$ or $(3,0)$.
However as @Guest points out in the comments you do need to careful about algebra. If you have $P=Q+R$ and $Q=R$, e.g. $Q$ is the amount of money a shopkeeper receives from you and $R$ is the amount of money you pay the shopkeeper ($Q$ is completely dependent on $R$), then $P=2Q=2R$. But if $Q$ and $R$ just follow the same distribution as in a Binomial variable being the sum of identically distributed Bernouilli variables, you have to keep the random variables separate: $X = Y_1 + Y_2 + \ldots + Y_n$ as you wrote (this is an example where the random variables being summed are independent).
