In Chapter 1's Problems, Problem 1(b), we are given that $X,Y\sim \beta_{1/2}:=\frac{1}{2}(\delta_0+\delta_1)$ are Bernoulli random variables. How am I to interpret this? That the probability of success is $\delta_0$ for each variable? Don't know what the $\frac{1}{2}$ is for.
Thanks!
$\beta_{\frac{1}{2}}=\frac{1}{2}(\delta_0+\delta_1)$ is a probability measure on the real numbers: $\beta_{\frac{1}{2}}(A)=1,\frac{1}{2},0$ depending on whether the set $A$ contains both of $0,1$, exactly one of $0,1$, or neither $0$ nor $1$. The factor $\frac{1}{2}$ is needed to make $\beta_{\frac{1}{2}}$ a probability measure.
Moreover, $\beta_{\frac{1}{2}}$ is the law of a Bernoulli random variable which takes the value $0$ with probability $\frac{1}{2}$ and takes the value $1$ with probability $\frac{1}{2}$.