# Tag Info

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The difference between the confidence interval for the mean response and the prediction interval is subtle but important. I'll explain it first and then provide you with a graphical intuition which helped me a lot when I learned this. Obviously, there is an error component to our prediction. Under the normal probability model, the prediction is normally ...

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$$\hat{\rho} \approx \frac{2\overline{x}}{\left(s_x^*\right)^2} +1$$ $$\Rightarrow \frac 12 (\hat{\rho} -1) \approx \frac{\overline{x}}{\left(s_x^*\right)^2} = [\left(s_x^*\right)^2]^{-1}\overline{x} = \Big(\frac{1}{T}\sum_{t=1}^T (x_t - \overline{x})^2\Big)^{-1}\cdot \Big(\frac{1}{T} \sum_{t=1}^T x_t\Big)$$ If we can assume that the process $\{X_t\}$ is ...

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You could integrate, substitute, and recognize a gamma function, but there is a nicer way to do it: recognize the pdf of a Gamma distribution. The pdf of a Gamma distribution is $$f(x) = \frac{\alpha^\beta}{\Gamma(\beta)} x^{\beta-1} e^{-\alpha x}, \qquad x \ge 0,$$ and its expectation is $\beta/\alpha$. So, in your case, we have a Gamma(2,$\alpha$), and ...

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In association rule learning, the confidence of a rule is defined as follows: $$conf(X\Rightarrow Y) = \frac{support(X\cup Y)}{support(X)}$$ The confidence is the amount of times a rule has been encountered in the data, conditional on the amount of times its left hand side was encountered. $100\%$ confidence implies that any record containing $X$ also ...

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This seems to be a somewhat strange design. It does not make much sense in an industrial setting: do you really want to generalize to the population of papers to compare the effect of two very specific pencils? You could not say anything about the pencil brand (unless the pencils of the brand are completely identical, but then the variance would have to ...

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$E(Y)$ is just the mean of your responses: it's the same thing as a regression where all you have is an intercept. For all values of $X$, you are predicting the same value for the response. We use conditional expectation because we expect there to be a relationship between a predictor variable and the response variable, such that we want our predictions to ...

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