Suppose the random variables $z_i $ are i.i.d. draw from the standard normal distribution $\mathcal{N}(0, 1)$, two vector $\mathbf{v}=(v_1, 0)^\top$, $v_1>0$, $\mathbf{w}=(w_1, w_2)^\top$, the inner angle between $\mathbf{v}$, $\mathbf{w}$ is $\theta$, $\operatorname{1}$ stands for indicator function (i.e. the its footnote condition is satisfied, return 1 else 0).
How to calculate $\mathbb{E}[z_1 \operatorname{1}_{z_1v1> 0, z_1w_1+z_2w_2>0}]$ ?
I guess this can be solved via integral over the feasible region ?