I want to find an expression for the variance of $\hat{b}=\frac{\sum_{t=1}^N\lambda^{N-t} y_t u_t}{\sum_{t=1}^{N}\lambda^{N-t}u_t^2}$ where $u_t$ is the deterministic input signal of the process $y_t=bu_t+e_t$ and $e_t$ is white Gaussian noise with variance $\sigma_e^2$. Also $\lambda$ is a constant value.
Im stuck since I dont know how to handle the summations when dealing with variance
One thing you need to clarify first is the following. Are $y_t=bu_t+e_t$ with $t=1,2,...,N$ uncorrelated? If it's the case you should know the general equation for sum of random variable 1. You only need to work with the upper term.