$$ f(x)= \begin{cases} \pi + (1-\pi) \ \text{CDF}(\hat{y},\hat{y}c; 0) & \text{if } y = 0\ (1-\pi)\ \text{PDF}(\hat{y},\hat{y}c) & \text{if} y>0 \end{cases} $$
$$Y \sim \pi \mathcal{N} ( \hat{y}_h , \hat{y}_h c) + (1-\pi) \delta_0$$
$$dif = \frac{\sum y}{n0} - \frac{\sum y}{ n1} = n1 \sum y0 - n0 \sum y1 = \sum y \ x \times n \ x $$
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