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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?

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    $\begingroup$ 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. $\endgroup$ Commented Jun 29, 2023 at 11:00
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    $\begingroup$ The distribution of $X$ is not determined by the information given, unless perhaps you are implicitly supposing the $X_i$ are independent. $\endgroup$
    – whuber
    Commented Jun 29, 2023 at 13:18

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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.

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