Distribution of vector of $n$ Bernoulli trials

I have a random variable $$X\in\{-1, 1\}^n$$ that is just a vector of length $$n$$ containing either $$-1$$ or $$+1$$. I know that $$\rho$$ is the probability of the event $$X = +1$$. I would like to describe the distribution of $$X$$.

What is the distribution of $$X$$?

The expression of the distribution for for $$x = (x_1, \ldots, x_n)$$ is $$\mu(dx) = \prod_{i=1}^n \rho\delta_{+1}(dx_i) + (1-\rho)\delta_{-1}(dx_i)$$ but what is the name of this distribution?

• It's a recoded Bernoulli distribution with $X=2Y -1$ where $Y$ has a Bernoulli($p$) distribution. I have encountered the name Rademacher($p$) distribution. Commented Jun 29, 2023 at 11:00
• The distribution of $X$ is not determined by the information given, unless perhaps you are implicitly supposing the $X_i$ are independent.
– whuber
Commented Jun 29, 2023 at 13:18

Assuming that $$X_i$$'s are independent, you got the joint distribution right. As for the name, not every distribution has a name. There are infinitely many possible distributions, we didn't name them all.