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Expected Value of Normal Distribution
$$\int \frac{\exp(\frac{(t - x)^2}{-2 \sigma^2})}{\sigma \sqrt{2 \pi}} \cdot t \cdot dt$$$$= \frac{\text{erf}(\frac{t-x}{\sigma \sqrt{2}})}{2} x - \frac{\sigma}{\pi \sqrt{2}} \exp(\frac{(t - x)^2}{-2 \sigma^2})$$