### Expected Value of Normal Distribution

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

### Expected Value of Normal Distribution

$$erf(-\infty) = -1$$ $$erf(0) = 0$$ $$erf(\infty) = 1$$