Expectation of random sum of non-random numbers I have a continuous random variable $\tau$ and I want to evaluate
$$
E\left(\sum_{i=1}^{\lfloor \tau \rfloor} Y_i\right),
$$
where $Y_i$ are known, non-random, and $\lfloor . \rfloor$ is the floor function. If $Y_i$s were iid I know I could use Wald's equation, for instance, but that is not the case. I am able to solve this through Monte Carlo, as I can simulate different $\tau$s. However, this will be very time-consuming since $Y_i$ can be big and the Monte Carlo samples can be large. It would be significantly easier if I could approximate the expectation above with
$$
\sum_{i=1}^{\lfloor E(\tau) \rfloor} Y_i.
$$
Is there a theoretical guarantee of this approximation?
Note on support: The vectors $Y_i$ are typically not large in magnitude, but they can be large in dimensions. The domain of $\tau$ is fixed to be $(1,N)$, where $N$ is known in advance, and it is unimodal.

This approximation will definitely not work in general. Consider a $\tau$ with support on $(1,3)$ and $E(\lfloor\tau\rfloor)\in [1,2)$. Then
$$
\sum_{i=1}^{E(\lfloor \tau \rfloor)} Y_i = Y_1,
$$
but if $Y_2\to\infty$ and $\tau$ has any mass at all on $[2,3)$,
$$
E\left(\sum_{i=1}^{\lfloor \tau \rfloor} Y_i\right)\to\infty.
$$
You will need to include more information on the distribution of $\lfloor\tau\rfloor$ and the whole vector $Y$.
