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I've found out that Rademacher random variables and Bernoulli random variables plays an important role in Probability theory. I am wondering how they are connected. For example,

Let $r_i, i=1, \ldots, n$ be Rademacher random variables, such that $P(r_i=1)=a$ and $P(r_i=-1)=1-a$.

Let $b_i, i=1, \ldots, n$ be Bernoulli random variables, such that $P(b_i=1)=b$ and $P(b_i=0)=1-b$.

Let $x_i, i=1, \ldots, n$ be real numbers.

Consider $A=\sum_{i=1}^nx_ir_i$.

How to represent $A$ in terms of $B=\sum_{i=1}^nx_ib_i$? What is the relation between $a$ and $b$?

Thank you.

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up vote 0 down vote accepted


If $r_i$ is a Rademacher random variable, then consider $\dfrac{r_i+1}{2}$.

Now consider $\displaystyle\sum_i x_i \dfrac{r_i+1}{2}$ or equivalently $\dfrac{\displaystyle\sum_i x_i r_i}{2} + \dfrac{\displaystyle\sum_i x_i }{2}$.

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