I'm homelearning statistics and trying to solve the following problem:
We have regression model $\tilde{y}=a \cdot \cos(x)+\varepsilon$ . Therefore $\tilde{y}_k = a \cdot \cos(x_k) + \varepsilon_k$, where $\varepsilon_k \approx N(0, \sigma^2), k= \overline{1,n} $.
Calculate an estimate of $\widehat{a}$ using the method of least squares. What is the expected value of $\tilde{y}_k$? What is the dispersion of $\tilde{y}_k$? Prove that $\widehat{a}$ is an unbiased estimate and show that the dispersion of $\widehat{a}=\frac{\sigma^2}{\sum_k^n(\cos^2(x_k))}$ .
$\varepsilon_k$ is just "noise", or a deviation.
This is my attempt, can you verify it whether it is correct please?
$\hat{a}=\frac{\sum \tilde{y}_i \cdot \cos(x_i)}{\sum \cos^2(x_i)}$
$E\tilde{y}_k=E(a\cos(x_i)+\epsilon_i)=a \cdot \cos(x_i)+E(\epsilon_i)=a \cdot \cos(x_i)$
$D\tilde{y}_i=D(a\cos(x_i)+\epsilon_i)=D(\epsilon_i)=\sigma^2$
$E(\hat{a})=\frac{1}{\sum \cos^2(x)x_i} \cdot E(\sum \tilde{y}_i \cdot \cos(x_i))=\frac{1}{\sum \cos^2(x_i)} \cdot \sum E(\tilde{y}_i \cdot \cos(x_i))=\frac{1}{\sum \cos^2(x_i)} \cdot E(\tilde{y}_i) = \frac{1}{\sum \cos^2(x_i)} \cdot \sum {\cos(x_i)} \cdot a \cdot \cos(x_i)=a \cdot \frac{\sum \cos^2(x_i)}{\sum \cos^2(xi)}=a$
$D(\hat{a})=D\frac{\sum \tilde{a}_i \cdot \cos(x_i)}{\sum \cos^2(x_i)}=\frac{1}{\sum \cos^2(x_i)} \cdot D(\sum \tilde{y_i} \cdot \cos(x_i))=(\frac{1}{\sum \cos^2(x_i)})^2 \cdot \sum D(\tilde{y}_i \cdot \cos(x_i))=(\frac{1}{\sum \cos^2(x_i)})^2 \cdot \sum \cos^2(x_i) \cdot D(\tilde{y}_i)=(\frac{1}{\sum \cos^2(x_1)})^2 \cdot \sum \cos^2(x_i) \cdot \sigma^2=\sigma^2 \cdot \frac{\sum \cos^2(x_i)}{(\sum \cos^2(x_i))^2}=\frac{\sigma^2}{\sum \cos^2 (x_i)}$
Is it correct? Thanks