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| Both sides previous revision Previous revision Next revision | Previous revision | ||
| kb:linear_regression [2022-03-02 02:35] – [Quantifying error] jaeyoung | kb:linear_regression [2024-04-30 04:03] (current) – external edit 127.0.0.1 | ||
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| The expectation is $0$, of course: | The expectation is $0$, of course: | ||
| + | $$ \mathbb{E}[\hat{Y}(x) - a^* - b^*x] = 0 $$ | ||
| + | |||
| + | The variance is: | ||
| + | |||
| + | $$ \mathrm{Var}[\hat{Y}(x) - a^* - b^*x] = \mathbb{E}[(\hat{Y}(x) - a^* - b^*x)^2] = \frac{\sigma^2}{n} \left( \frac{(x - \bar{x})^2}{\sigma_x^2} \frac{n-1}{n} + 1 \right) $$ | ||
| + | |||
| + | The distribution is Gaussian if $\varepsilon_i$ are Gaussian. If it is Gaussian, then we can easily compute [[kb: | ||
| ===== Multivariate linear regression ===== | ===== Multivariate linear regression ===== | ||