Homework

This is homework (1) 已知： $$X_1, X_2, …, X_n \overset{\text{iid}}{\sim}p(x)$$ 計算： $$ E( \hat{I}_M)=E\left[\frac{1}{n} \sum^n_{i=1} \frac{f(X_i)}{p(X_i)} \right]=\frac{1}{n}E\left[ \sum^n_{i=1} \frac{f(X_i)}{p(X_i)} \right] $$ 對於每個獨立的 $X_i$ ，我們只要計算： $$E\left[\frac{f(X_i)}{p(X_i)} \right]$$ 因此： $$ E\left[\frac{f(X)}{p(X)} \right] = \int^b_a\frac{f(x)}{p(x)}p(x)dx =\int^b_af(x)dx = I $$ 可知 $$E\left[\frac{f(X_i)}{p(X_i)} \right] =I,　\forall i $$ 所以 $$ E(\hat{I}_M) =\frac{1}{n}\sum^n_{i=1}I=I $$ (2) 計算變異數 $$Var(\hat{I}_M)=E\left[(\hat{I}_M-I)^2\right]$$ 因為 $$ \begin{aligned} Var(\widehat{I}_M) &= Var\left(\frac{1}{n} \sum_{i=1}^{n} \frac{f(X_i)}{p(X_i)}\right) = \frac{1}{n}Var\left(\frac{f(X)}{p(X)}\right) \\ &= \frac{1}{n}\left(E\left[\left(\frac{f(X)}{p(X)}\right)^2\right]-I^2\right) \end{aligned} $$ 已知 $$E\left[\left(\frac{f(X)}{p(X)}\right)^2\right] < \infty$$ ...

Assume $$ Y_i = \beta_o + \beta_1X_i+\varepsilon_i $$ , for given $n$ observerd data $(x_i, Y_i)$, $\forall i=1～n$ Note that: $$ Y_i \mid X_i=x_i ~ N(\beta_o+\beta_1X_1, \sigma^2) $$ $\therefore$ $$ E_{Y \mid X}[Y_i \mid X_i =x_i]=\beta_o+\beta_1X_1$$ In vector notation: $$ Y_i = x^T_i\beta + \varepsilon_i $$ where $ x_i=(1, X_1, …, X_n)^T $ And for $ Y = (Y_1, …, Y_n)^T $, We have: $$ Y = X\beta + \varepsilon $$ ...