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.

Set $S_t = \sum_{i=1}^t Y_i$. Then
$$
E[S_\tau\mid \tau \leq T] = \sum_{t=1}^T S_t P(\tau = t) \triangleq H_T
=H_{T-1} + S_T\cdot P(\tau = T)
$$
Y = getY()
p = simulatep()
H = 0
T = -1
S = 0
while not converged:
    T += 1
    S = S + Y[T]
    H = H + S * p[T]

