A joint demosaicing and denoising task refers to the task of simultaneously reconstructing and denoising a color image from a patterned image obtained by a monochrome image sensor with a color filter array. Recently, inspired by the success of deep learning in many image processing tasks, there has been research to apply convolutional neural networks (CNNs) to the task of joint demosaicing and denoising. However, such CNNs need many training data to be trained, and work well only for patterned images which have the same amount of noise they have been trained on. In this paper, we propose a variational deep image prior network for joint demosaicing and denoising which can be trained on a single patterned image and works for patterned images with different levels of noise. We also propose a new RGB color filter array (CFA) which works better with the proposed network than the conventional Bayer CFA. Mathematical justifications of why the variational deep image prior network suits the task of joint demosaicing and denoising are also given, and experimental results verify the performance of the proposed method.