Consider the regression model $Y=\beta_1 X_1+\beta_2 X_2+\epsilon$, with a sample of size $n$, $Y_i=\beta_1 X_{i1}+\beta_2 X_{i2}+\epsilon_i$, $\epsilon_i \in N(0,\sigma^2)$. Suppossing $\bar{X_1}=\bar{X_2}=0$ and $X_1,X_2$ orthogonal, i.e., $\sum\limits_{i=1}^n X_{i1} X_{i2}=0$.
Is there any relation between the Coefficient of Multiple Correlation and the Coefficients of Simple Correlation between $Y$ and $X_1$ and between $Y$ and $X_2$.
I got to:
$$r(Y;X_1,X_2)=\dfrac{\hat{\beta_1}\sum\limits_{i=1}^n Y_i X_{i1} + \hat{\beta_2}\sum\limits_{i=1}^n Y_i X_{i2}}{\sqrt{\sum\limits_{i=1}^n (Y_i-\bar{Y})^2 \left[ \hat{\beta_1}\sum\limits_{i=1}^n X_{i1}^2 + \hat{\beta_2}\sum\limits_{i=1}^n X_{i2}^2\right]}}$$
$$r(Y,X_1)=\dfrac{\sum\limits_{i=1}^n Y_i X_{i1}}{\sqrt{\sum\limits_{i=1}^n (Y_i-\bar{Y})^2 \sum\limits_{i=1}^n X_{i1}^2}}$$
$$r(Y,X_2)=\dfrac{\sum\limits_{i=1}^n Y_i X_{i2}}{\sqrt{\sum\limits_{i=1}^n (Y_i-\bar{Y})^2 \sum\limits_{i=1}^n X_{i2}^2}}$$
But I cannot find the relation