Showing that $\hat{\beta}_1$ can be expressed as a sum I'm feeling very uncertain about my method of showing that for linear regression, $$\hat{\beta}_1 = \sum_{i=1}^n k_i Y_i$$ where $k_i = \frac{x_i-\bar{x}}{S_{xx}}$. Basically, is my following method correct?
$$\hat{\beta}_1 = \frac{S_{xy}}{S_{xx}} = \frac{1}{S_{xx}}\sum_{i=1}^n (x_i - \bar{x})Y_i = \sum_{i=1}^n \left(\frac{x_i - \bar{x}}{S_{xx}}\right)Y_i = \sum_{i=1}^n k_iY_i$$
Part of the reason why I'm very confused is that $S_{xy}$ has a lower case $y$ and the sum itself has an upper case $Y$, which I thought meant different things (I believe $y$ denotes an actual data value while $Y$ is a random variable).
If anyone is wondering, I had already shown that $$S_{xy} = \sum_{i=1}^n (x_i - \bar{x})y_i$$ which I noted has a different case for $y$ then what the original question had asked, again adding to my confusion.
 A: The distinction between lower-case $y$ and upper-case $Y$ is used to distinguish the observed data (treated as constant) from the unobserved random variable.  The same distinction applies to the estimator, which can either be a random variable, or an observed constant (an estimate):
$$\begin{aligned}
\text{Estimator} & & & \hat{\beta}_1(\boldsymbol{Y}) = \frac{S_{xY}}{S_{xx}} = \sum_{i=1}^n k_i Y_i, \\[10pt]
\text{Estimate} & & & \text{ } \hat{\beta}_1(\boldsymbol{y}) = \frac{S_{xy}}{S_{xx}} = \sum_{i=1}^n k_i y_i. \\[10pt]
\end{aligned}$$
A: I see your confusion, and you're right about the letters in general. In probability courses, you commonly see something like $P(X=x)$. Here, $X$ refers to the RV, and $x$ refers to a specific value. This convention is also followed in statistics. But, it is messier compared to a prob. course :), so you can sometimes have slight notation diversions. 
For your case, when you have real data, and you want to estimate $\hat{\beta}_1$, you can substitute your values instead of $Y_i$, thinking of it like $y_i$. But, this is a formula and you can also think of these $y_i$ as $Y_i$ and calculate the expected value and variance of $\hat{\beta}_1$, as if it is a random variable. It's a matter of perspective. 
