The mean function of $Y_t$ is calculated as follows:

$\mu_Y(t) = E(Y_t) = E(1 + 0.5t + Z_t)$

Since $1 + 0.5t$is deterministic (non-random), and$Z_t \sim WN(0,1)$where$WN$ stands for White Noise with mean 0 and variance 1:

$\mu_Y(t) = 1 + 0.5t + E(Z_t)$

Given that $E(Z_t) = 0$:

$\mu_Y(t) = 1 + 0.5t$

Therefore, the correct answer is C: $1 + 0.5t$.