# When does replacing one random variable with another not change the joint distribution

Suppose $X_1, \dots, X_n$ are random variables with some joint distribution.

I wonder if there is some concept or condition for that if replacing $X_i$ with some other random variable $Y$, then $X_1, \dots, X_{i-1}, Y, X_{i+1}, \dots, X_n$ still have the same joint distribution?

One condition I know is that when $X_1, \dots, X_n, Y$ are independent, and $X_i$ and $Y$ are identically distributed. I wonder if there are weaker conditions?

Thanks!

## 1 Answer

I think what you're looking for is that $X_1, \ldots, X_n, Y$ are exchangeable.

• Thanks! Yes, it is one sufficient condition. I did think of that, but forgot to mention it in my question. Note that exchangebility isn't weaker than "X1,…,Xn,Y are independent, and Xi and Y are identically distributed." Both conditions may be stronger than necessary. – Tim Oct 19 '12 at 0:36