Linear Separability

Let $$({{X}_{i}},{{\theta }_{i}}), i = 1,2, \ldots ,n$$, be i.i.d. random pairs, where {θ_i} is Bernoulli with parameter 1/2, and X_i ~ f_θi(x), x_i ∈ Rd. We say $$({{X}_{i}},{{\theta }_{i}}), i = 1,2, \ldots ,n$$ is linearly separable if there exits a vector w ∈ Rd and a real number T such that $$\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{{w}^{t}}{{x}_{i}} \geqslant T,} & {{{\theta }_{i}} = 1} \\ \end{array} } \hfill \\ {\begin{array}{*{20}{c}} { < T,} & {{{\theta }_{i}} = 0,} & {for} & {i = 1,2, \ldots ,n.} \\ \end{array} } \hfill \\ \end{array}$$