 Solution

Put $z_i=e^{-t_i}-c$. Assume the measurement errors are independent and centered, so $Y_i=\alpha +\beta z_i+{ϵ}_i$ with independent ${ϵ}_i\sim N(0,{\sigma }^2)$. Treat the observation times as fixed. The design matrix has rows $(1,z_i)$ and parameter vector $(\alpha ,\beta {)}^{\mathsf{T}}$. Require at least two distinct times so its columns are independent, equivalently ${\sum }_i{z}_i^2>0$. This gives the required Gaussian linear regression model.