Printed Electronics (PE) is a promising technology that provides mechanical flexibility and low-cost fabrication. These features make the PE the key enabler of the emerging applications, such as smart sensors, wearables, and Internet of Things (IoTs). Since these applications need secure communication and/or authentication, it is vital to utilize security primitives for cryptographic key and identification. Physical Unclonable Function (PUF) has been adopted widely to provide the secure keys. In this work, we present a weak PUF based on Electrolyte-gated FETs using inorganic inkjet printed electronics. A comprehensive analysis framework including Monte Carlo simulations based on real device measurements is developed to evaluate the proposed PE-PUF. Moreover, the multi-bit PE-PUF design is optimized to reduce the area usage. The analysis results show that the PE-PUF has ideal uniqueness, sufficient reliability, and operates at low voltage (1 V) which is critical for the PE applications. In addition, the optimization of the multi-bit PE-PUF reduces the area usage around 30%.