# 回归分析|Regression Analysis代写 STAT 525

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Population regression function, or simply, the regression function:
$$\mu_{Y}(x)=\beta_{0}+\beta_{1} x \quad \text { for } a \leq x \leq b$$
Sample regression function:
$$\hat{\mu}{Y}(x)=\hat{\beta}{0}+\hat{\beta}{1} x$$ Population regression model, or simply, the regression model: $$Y{I}=\beta_{0}+\beta_{1} X_{I}+E_{I} \quad \text { for } I=1, \ldots, N$$

Sample regression model:
$$y_{i}=\beta_{0}+\beta_{1} x_{i}+e_{i} \quad \text { for } i=1, \ldots, n$$
A randomly chosen $Y$ value from the subpopulation determined by $X=x$ :
$$\boldsymbol{Y}(\boldsymbol{x})$$
Sample prediction function, or simply, prediction function:
$$\hat{Y}(x)=\hat{\beta}{0}+\hat{\boldsymbol{\beta}}{1} \boldsymbol{x}$$
Note: $\hat{\mu}_{Y}(x)=\hat{Y}(x)$

when $X$ and $Y$ are measured using the first system of units. Also suppose the regression function is
$$\mu_{Y^{}}\left(x^{}\right)=\beta_{0}^{}+\beta_{1}^{} x^{}$$ when $X^{}$ and $Y^{}$ are measured using the second system of units. Then it can be proved mathematically that $$\beta_{1}^{}=\frac{d}{b} \beta_{1}$$
$$\beta_{0}^{*}=c+\frac{d}{b}\left(b \beta_{0}-a \beta_{1}\right)$$