I have a time-varying signal $f(t)$. Couple of probes are trying to measure its values across different time intervals:
\begin{align} X_1 &= f(t = t_0) + e_1 \\ X_2 &= f(t = t_0+h) + e_2 \end{align}
$e_1$ and $e_2$ are two independent random variables with zero mean and variance = $\sigma^2$. Goal is to estimate the rate of change of the signal using following estimator
$$R = (X_2 - X_1)/h$$
How should I calculate $E(R)$ and $\text{Var}(R)$?
Unless I'm missing something, you're overthinking this problem. The expectation operator is linear, so we have: $$ E(R) = \frac{1}{h} \left( E(X_1) - E(X_2) \right) $$
Then all you have to do is plug in some numbers.
Edit: $f$ is a known function and $t_0$ and $h$ are known values, so $E\left(f(t=t_0)\right)=f(t=t_0)$ and $E\left(f(t=t_0+h)\right)=f(t=t_0+h)$. Then by the same linearity used above, we have $$ E\left(f(t=t_0) + e_1\right) = E\left(f(t=t_0)\right) + E\left(e_1\right) $$ and $$ E\left(f(t=t_0 + h) + e_2\right) = E\left(f(t=t_0 + h)\right) + E\left(e_2\right). $$
Since $E(e_1) = E(e_2) = 0$ by assumption, the expectations are just $f(t=t_0)$ and $f(t=t_0+h)$. Then $E(R) = \frac{1}{h}(f(t=t_0) - f(t=t_0+h))$.
