Many biological processes are controlled by biomolecular switches which themselves are regulated by various upstream chemical molecules (the input). Understanding how input noise affects the output stochastic switching process is of significant interest in various biophysical systems like gene regulation, chemosensing, and cell motility. Here, we propose an exactly solvable model where the noisy input signal arises from a simple birth-death process and directly regulates the transition rates of a downstream switch. We solve the joint master equations to analyze the statistical properties of the output switching process. Our results suggest that the conventional wisdom of an additive input-output noise rule fails to describe signaling systems containing a single molecular switch, and, instead, the most important effect of input noise is to effectively reduce the on rate of the switch.