The following distribution came up in my research and I'm wondering if the general class has a name and if there are computationally efficient sampling methods for it.
Let $\mathcal{X} = \{0,1\}^{n}$ be the sample space; these are simply $n$ dimensional vectors with each component either $0$ or $1$. Let $a_1,...,a_n$ be non-negative coefficients with atleast one $a_i$ positive. Then define the measure $$P(x_1,...,x_n) = \frac{1}{Z}\sum\limits_{i=1}^{n}a_ix_i$$ where $Z$ is a normalization constant (a simple recursion shows $Z = 2^{n-1}\sum\limits_{i=1}^{n}a_i)$.
Does this distribution look familiar to anyone? Please include a reference to a textbook or scholarly article in your response.
